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Properties

Label 10-1045e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.246\times 10^{15}$
Sign $-1$
Analytic cond. $40454.5$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 7·3-s + 2·4-s + 5·5-s + 21·6-s − 11·7-s + 6·8-s + 21·9-s − 15·10-s − 5·11-s − 14·12-s + 13-s + 33·14-s − 35·15-s − 17·16-s − 3·17-s − 63·18-s + 5·19-s + 10·20-s + 77·21-s + 15·22-s − 8·23-s − 42·24-s + 15·25-s − 3·26-s − 29·27-s − 22·28-s + ⋯
L(s)  = 1  − 2.12·2-s − 4.04·3-s + 4-s + 2.23·5-s + 8.57·6-s − 4.15·7-s + 2.12·8-s + 7·9-s − 4.74·10-s − 1.50·11-s − 4.04·12-s + 0.277·13-s + 8.81·14-s − 9.03·15-s − 4.25·16-s − 0.727·17-s − 14.8·18-s + 1.14·19-s + 2.23·20-s + 16.8·21-s + 3.19·22-s − 1.66·23-s − 8.57·24-s + 3·25-s − 0.588·26-s − 5.58·27-s − 4.15·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 11^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 11^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{5} \cdot 11^{5} \cdot 19^{5}\)
Sign: $-1$
Analytic conductor: \(40454.5\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 5^{5} \cdot 11^{5} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$ \( ( 1 - T )^{5} \)
11$C_1$ \( ( 1 + T )^{5} \)
19$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \times (C_2^4 : C_5)$ \( 1 + 3 T + 7 T^{2} + 9 T^{3} + 3 p^{2} T^{4} + 13 T^{5} + 3 p^{3} T^{6} + 9 p^{2} T^{7} + 7 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
3$C_2 \times (C_2^4 : C_5)$ \( 1 + 7 T + 28 T^{2} + 26 p T^{3} + 172 T^{4} + 319 T^{5} + 172 p T^{6} + 26 p^{3} T^{7} + 28 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \times (C_2^4 : C_5)$ \( 1 + 11 T + 68 T^{2} + 286 T^{3} + 136 p T^{4} + 2673 T^{5} + 136 p^{2} T^{6} + 286 p^{2} T^{7} + 68 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \times (C_2^4 : C_5)$ \( 1 - T + 28 T^{2} - 5 T^{3} + 569 T^{4} - 321 T^{5} + 569 p T^{6} - 5 p^{2} T^{7} + 28 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \times (C_2^4 : C_5)$ \( 1 + 3 T + 60 T^{2} + 145 T^{3} + 1605 T^{4} + 3219 T^{5} + 1605 p T^{6} + 145 p^{2} T^{7} + 60 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \times (C_2^4 : C_5)$ \( 1 + 8 T + 123 T^{2} + 707 T^{3} + 5854 T^{4} + 24057 T^{5} + 5854 p T^{6} + 707 p^{2} T^{7} + 123 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \times (C_2^4 : C_5)$ \( 1 - 11 T + 112 T^{2} - 627 T^{3} + 3559 T^{4} - 16423 T^{5} + 3559 p T^{6} - 627 p^{2} T^{7} + 112 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \times (C_2^4 : C_5)$ \( 1 + 5 T + 44 T^{2} + 25 T^{3} + 1349 T^{4} + 2589 T^{5} + 1349 p T^{6} + 25 p^{2} T^{7} + 44 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \times (C_2^4 : C_5)$ \( 1 + 9 T + 147 T^{2} + 894 T^{3} + 257 p T^{4} + 45431 T^{5} + 257 p^{2} T^{6} + 894 p^{2} T^{7} + 147 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \times (C_2^4 : C_5)$ \( 1 - 15 T + 251 T^{2} - 2279 T^{3} + 21553 T^{4} - 135855 T^{5} + 21553 p T^{6} - 2279 p^{2} T^{7} + 251 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \times (C_2^4 : C_5)$ \( 1 + 13 T + 100 T^{2} + 452 T^{3} + 2900 T^{4} + 19551 T^{5} + 2900 p T^{6} + 452 p^{2} T^{7} + 100 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \times (C_2^4 : C_5)$ \( 1 + 20 T + 307 T^{2} + 3135 T^{3} + 28704 T^{4} + 205341 T^{5} + 28704 p T^{6} + 3135 p^{2} T^{7} + 307 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \times (C_2^4 : C_5)$ \( 1 + 5 T + 110 T^{2} + 377 T^{3} + 79 p T^{4} + 15789 T^{5} + 79 p^{2} T^{6} + 377 p^{2} T^{7} + 110 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \times (C_2^4 : C_5)$ \( 1 + 17 T + 360 T^{2} + 3969 T^{3} + 46067 T^{4} + 350185 T^{5} + 46067 p T^{6} + 3969 p^{2} T^{7} + 360 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \times (C_2^4 : C_5)$ \( 1 - 3 T + 258 T^{2} - 673 T^{3} + 28951 T^{4} - 59539 T^{5} + 28951 p T^{6} - 673 p^{2} T^{7} + 258 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \times (C_2^4 : C_5)$ \( 1 + 28 T + 510 T^{2} + 6438 T^{3} + 68729 T^{4} + 599427 T^{5} + 68729 p T^{6} + 6438 p^{2} T^{7} + 510 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \times (C_2^4 : C_5)$ \( 1 + 6 T + 343 T^{2} + 1595 T^{3} + 47738 T^{4} + 165975 T^{5} + 47738 p T^{6} + 1595 p^{2} T^{7} + 343 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \times (C_2^4 : C_5)$ \( 1 + 16 T + 452 T^{2} + 4858 T^{3} + 72491 T^{4} + 538763 T^{5} + 72491 p T^{6} + 4858 p^{2} T^{7} + 452 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \times (C_2^4 : C_5)$ \( 1 - 3 T + 249 T^{2} - 526 T^{3} + 32913 T^{4} - 60865 T^{5} + 32913 p T^{6} - 526 p^{2} T^{7} + 249 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \times (C_2^4 : C_5)$ \( 1 + 33 T + 756 T^{2} + 12067 T^{3} + 154591 T^{4} + 1547711 T^{5} + 154591 p T^{6} + 12067 p^{2} T^{7} + 756 p^{3} T^{8} + 33 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \times (C_2^4 : C_5)$ \( 1 + 16 T + 334 T^{2} + 4254 T^{3} + 55881 T^{4} + 513531 T^{5} + 55881 p T^{6} + 4254 p^{2} T^{7} + 334 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \times (C_2^4 : C_5)$ \( 1 + 14 T + 460 T^{2} + 4966 T^{3} + 86849 T^{4} + 703275 T^{5} + 86849 p T^{6} + 4966 p^{2} T^{7} + 460 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.36370934179019048522105382055, −6.28881268833515668189081762988, −6.14578308566752869084308129119, −5.99488277320995693797376354037, −5.78538816656008188925923333597, −5.72780243667146431756540116883, −5.37753319588835800476599691347, −5.35707799224462277849597441460, −5.27799974047905390366857456996, −4.90727779567080300743424068860, −4.77236175693271245142212910116, −4.50551575743991790104397101702, −4.33510243722719969775879883865, −4.27756842017052577651789645402, −3.74060626740907007332681735399, −3.42197831029460537931350324079, −3.19789646954358869850323480873, −3.03566775091113100305494993993, −2.75598085987582934371435161871, −2.60532323274751192430795141756, −2.47822128914369870164110130036, −1.72958771278925189564283111174, −1.46912671416111265120393175103, −1.28254953411861682805164698558, −1.15756315766356415567357572527, 0, 0, 0, 0, 0, 1.15756315766356415567357572527, 1.28254953411861682805164698558, 1.46912671416111265120393175103, 1.72958771278925189564283111174, 2.47822128914369870164110130036, 2.60532323274751192430795141756, 2.75598085987582934371435161871, 3.03566775091113100305494993993, 3.19789646954358869850323480873, 3.42197831029460537931350324079, 3.74060626740907007332681735399, 4.27756842017052577651789645402, 4.33510243722719969775879883865, 4.50551575743991790104397101702, 4.77236175693271245142212910116, 4.90727779567080300743424068860, 5.27799974047905390366857456996, 5.35707799224462277849597441460, 5.37753319588835800476599691347, 5.72780243667146431756540116883, 5.78538816656008188925923333597, 5.99488277320995693797376354037, 6.14578308566752869084308129119, 6.28881268833515668189081762988, 6.36370934179019048522105382055

Graph of the $Z$-function along the critical line