Properties

Label 10-6027e5-1.1-c1e5-0-2
Degree $10$
Conductor $7.953\times 10^{18}$
Sign $-1$
Analytic cond. $2.58161\times 10^{8}$
Root an. cond. $6.93727$
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 + 5·3-s + 3·4-s − 5-s − 15·6-s − 8-s + 15·9-s + 3·10-s + 4·11-s + 15·12-s − 3·13-s − 5·15-s − 2·16-s + 8·17-s − 45·18-s − 20·19-s − 3·20-s − 12·22-s − 5·23-s − 5·24-s − 10·25-s + 9·26-s + 35·27-s + 9·29-s + 15·30-s − 16·31-s + 6·32-s + ⋯
L(s)  = 1  − 2.12·2-s + 2.88·3-s + 3/2·4-s − 0.447·5-s − 6.12·6-s − 0.353·8-s + 5·9-s + 0.948·10-s + 1.20·11-s + 4.33·12-s − 0.832·13-s − 1.29·15-s − 1/2·16-s + 1.94·17-s − 10.6·18-s − 4.58·19-s − 0.670·20-s − 2.55·22-s − 1.04·23-s − 1.02·24-s − 2·25-s + 1.76·26-s + 6.73·27-s + 1.67·29-s + 2.73·30-s − 2.87·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 7^{10} \cdot 41^{5}\)
Sign: $-1$
Analytic conductor: \(2.58161\times 10^{8}\)
Root analytic conductor: \(6.93727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{5} \cdot 7^{10} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3$C_1$ \( ( 1 - T )^{5} \)
7 \( 1 \)
41$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + 3 T + 3 p T^{2} + 5 p T^{3} + 17 T^{4} + 27 T^{5} + 17 p T^{6} + 5 p^{3} T^{7} + 3 p^{4} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + T + 11 T^{2} + 18 T^{3} + 93 T^{4} + 19 p T^{5} + 93 p T^{6} + 18 p^{2} T^{7} + 11 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 4 T + 31 T^{2} - 104 T^{3} + 482 T^{4} - 1400 T^{5} + 482 p T^{6} - 104 p^{2} T^{7} + 31 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 3 T + 47 T^{2} + 10 p T^{3} + 83 p T^{4} + 2365 T^{5} + 83 p^{2} T^{6} + 10 p^{3} T^{7} + 47 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 8 T + 49 T^{2} - 300 T^{3} + 1438 T^{4} - 5452 T^{5} + 1438 p T^{6} - 300 p^{2} T^{7} + 49 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 20 T + 225 T^{2} + 1740 T^{3} + 10388 T^{4} + 49876 T^{5} + 10388 p T^{6} + 1740 p^{2} T^{7} + 225 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 5 T + 67 T^{2} + 190 T^{3} + 1867 T^{4} + 3751 T^{5} + 1867 p T^{6} + 190 p^{2} T^{7} + 67 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 9 T + 167 T^{2} - 1050 T^{3} + 10313 T^{4} - 45757 T^{5} + 10313 p T^{6} - 1050 p^{2} T^{7} + 167 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 16 T + 177 T^{2} + 1396 T^{3} + 320 p T^{4} + 58252 T^{5} + 320 p^{2} T^{6} + 1396 p^{2} T^{7} + 177 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 19 T + 263 T^{2} + 2510 T^{3} + 20673 T^{4} + 134497 T^{5} + 20673 p T^{6} + 2510 p^{2} T^{7} + 263 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 6 T + 51 T^{2} - 596 T^{3} + 4246 T^{4} - 368 p T^{5} + 4246 p T^{6} - 596 p^{2} T^{7} + 51 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 9 T + 139 T^{2} + 1184 T^{3} + 11621 T^{4} + 68361 T^{5} + 11621 p T^{6} + 1184 p^{2} T^{7} + 139 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 29 T + 471 T^{2} + 4914 T^{3} + 40351 T^{4} + 292431 T^{5} + 40351 p T^{6} + 4914 p^{2} T^{7} + 471 p^{3} T^{8} + 29 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 28 T + 407 T^{2} + 4136 T^{3} + 34394 T^{4} + 264148 T^{5} + 34394 p T^{6} + 4136 p^{2} T^{7} + 407 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 16 T + 231 T^{2} + 2468 T^{3} + 23928 T^{4} + 189756 T^{5} + 23928 p T^{6} + 2468 p^{2} T^{7} + 231 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 21 T + 435 T^{2} - 5528 T^{3} + 64669 T^{4} - 552269 T^{5} + 64669 p T^{6} - 5528 p^{2} T^{7} + 435 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 6 T + 293 T^{2} - 1568 T^{3} + 38320 T^{4} - 161040 T^{5} + 38320 p T^{6} - 1568 p^{2} T^{7} + 293 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 4 T + 179 T^{2} - 32 T^{3} + 13708 T^{4} + 37428 T^{5} + 13708 p T^{6} - 32 p^{2} T^{7} + 179 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 21 T + 431 T^{2} - 6172 T^{3} + 71153 T^{4} - 713941 T^{5} + 71153 p T^{6} - 6172 p^{2} T^{7} + 431 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 26 T + 519 T^{2} + 7688 T^{3} + 92050 T^{4} + 928764 T^{5} + 92050 p T^{6} + 7688 p^{2} T^{7} + 519 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 12 T + 385 T^{2} + 3752 T^{3} + 65750 T^{4} + 475176 T^{5} + 65750 p T^{6} + 3752 p^{2} T^{7} + 385 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 37 T + 959 T^{2} + 16574 T^{3} + 232537 T^{4} + 2506545 T^{5} + 232537 p T^{6} + 16574 p^{2} T^{7} + 959 p^{3} T^{8} + 37 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.04089288082352499718344106033, −4.97671706799637354831098402028, −4.75327475448378933664557451784, −4.56876581656094117165526408587, −4.32182495242022768080354246602, −4.26553093892515691862382657559, −4.02451743584651505199224332755, −4.00525012819142244934334251391, −3.86643290090145403394079503915, −3.66377107128757254063936895325, −3.56625091126591932571619040989, −3.32927252960867034938393734515, −3.26293190025188399995831565868, −2.94093767617984846768693039348, −2.90417848342372436914034124819, −2.64218604165871862023141742346, −2.38084734185006325142026297890, −2.21749598562916639504175686277, −2.04386199562584031441168428796, −1.99236295527835999941724038135, −1.67193689224286914883052457432, −1.60094051065918530099652698428, −1.40242223518632795744450711281, −1.21733912357512425717582194837, −1.13969407568616553876902409500, 0, 0, 0, 0, 0, 1.13969407568616553876902409500, 1.21733912357512425717582194837, 1.40242223518632795744450711281, 1.60094051065918530099652698428, 1.67193689224286914883052457432, 1.99236295527835999941724038135, 2.04386199562584031441168428796, 2.21749598562916639504175686277, 2.38084734185006325142026297890, 2.64218604165871862023141742346, 2.90417848342372436914034124819, 2.94093767617984846768693039348, 3.26293190025188399995831565868, 3.32927252960867034938393734515, 3.56625091126591932571619040989, 3.66377107128757254063936895325, 3.86643290090145403394079503915, 4.00525012819142244934334251391, 4.02451743584651505199224332755, 4.26553093892515691862382657559, 4.32182495242022768080354246602, 4.56876581656094117165526408587, 4.75327475448378933664557451784, 4.97671706799637354831098402028, 5.04089288082352499718344106033

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