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Properties

Label 10-4235e5-1.1-c1e5-0-7
Degree $10$
Conductor $1.362\times 10^{18}$
Sign $-1$
Analytic cond. $4.42234\times 10^{7}$
Root an. cond. $5.81520$
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  + 2·3-s − 3·4-s + 5·5-s − 5·7-s − 2·8-s − 2·9-s − 6·12-s − 6·13-s + 10·15-s + 16-s − 2·17-s − 7·19-s − 15·20-s − 10·21-s + 5·23-s − 4·24-s + 15·25-s − 11·27-s + 15·28-s − 11·29-s − 2·31-s + 7·32-s − 25·35-s + 6·36-s + 2·37-s − 12·39-s − 10·40-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s + 2.23·5-s − 1.88·7-s − 0.707·8-s − 2/3·9-s − 1.73·12-s − 1.66·13-s + 2.58·15-s + 1/4·16-s − 0.485·17-s − 1.60·19-s − 3.35·20-s − 2.18·21-s + 1.04·23-s − 0.816·24-s + 3·25-s − 2.11·27-s + 2.83·28-s − 2.04·29-s − 0.359·31-s + 1.23·32-s − 4.22·35-s + 36-s + 0.328·37-s − 1.92·39-s − 1.58·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{5} \cdot 7^{5} \cdot 11^{10}\)
Sign: $-1$
Analytic conductor: \(4.42234\times 10^{7}\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 5^{5} \cdot 7^{5} \cdot 11^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 - T )^{5} \)
7$C_1$ \( ( 1 + T )^{5} \)
11 \( 1 \)
good2$C_2 \wr S_5$ \( 1 + 3 T^{2} + p T^{3} + p^{3} T^{4} + 5 T^{5} + p^{4} T^{6} + p^{3} T^{7} + 3 p^{3} T^{8} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 2 T + 2 p T^{2} - 5 T^{3} + 10 T^{4} + 5 T^{5} + 10 p T^{6} - 5 p^{2} T^{7} + 2 p^{4} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 6 T + 57 T^{2} + 193 T^{3} + 86 p T^{4} + 217 p T^{5} + 86 p^{2} T^{6} + 193 p^{2} T^{7} + 57 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 2 T + 57 T^{2} + 13 T^{3} + 76 p T^{4} - 789 T^{5} + 76 p^{2} T^{6} + 13 p^{2} T^{7} + 57 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 7 T + 94 T^{2} + 476 T^{3} + 3562 T^{4} + 13107 T^{5} + 3562 p T^{6} + 476 p^{2} T^{7} + 94 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 5 T + 81 T^{2} - 237 T^{3} + 2605 T^{4} - 5531 T^{5} + 2605 p T^{6} - 237 p^{2} T^{7} + 81 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 11 T + 129 T^{2} + 927 T^{3} + 6775 T^{4} + 35345 T^{5} + 6775 p T^{6} + 927 p^{2} T^{7} + 129 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 2 T + 66 T^{2} + 67 T^{3} + 2646 T^{4} - 17 T^{5} + 2646 p T^{6} + 67 p^{2} T^{7} + 66 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 2 T + 46 T^{2} + 233 T^{3} + 912 T^{4} + 12917 T^{5} + 912 p T^{6} + 233 p^{2} T^{7} + 46 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 13 T + 221 T^{2} + 1723 T^{3} + 16843 T^{4} + 95105 T^{5} + 16843 p T^{6} + 1723 p^{2} T^{7} + 221 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 10 T + 92 T^{2} - 41 T^{3} - 2702 T^{4} - 41471 T^{5} - 2702 p T^{6} - 41 p^{2} T^{7} + 92 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 2 T + 108 T^{2} - 66 T^{3} + 7381 T^{4} - 2431 T^{5} + 7381 p T^{6} - 66 p^{2} T^{7} + 108 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 14 T + 310 T^{2} + 2901 T^{3} + 34832 T^{4} + 228545 T^{5} + 34832 p T^{6} + 2901 p^{2} T^{7} + 310 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 6 T + 195 T^{2} - 853 T^{3} + 17750 T^{4} - 58801 T^{5} + 17750 p T^{6} - 853 p^{2} T^{7} + 195 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 20 T + 454 T^{2} + 5390 T^{3} + 65255 T^{4} + 509133 T^{5} + 65255 p T^{6} + 5390 p^{2} T^{7} + 454 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 9 T + 116 T^{2} + 26 T^{3} + 1910 T^{4} - 33641 T^{5} + 1910 p T^{6} + 26 p^{2} T^{7} + 116 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 10 T + 4 p T^{2} + 2419 T^{3} + 36478 T^{4} + 246431 T^{5} + 36478 p T^{6} + 2419 p^{2} T^{7} + 4 p^{4} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 15 T + 378 T^{2} + 3994 T^{3} + 55070 T^{4} + 423151 T^{5} + 55070 p T^{6} + 3994 p^{2} T^{7} + 378 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 23 T + 372 T^{2} + 4656 T^{3} + 52204 T^{4} + 490407 T^{5} + 52204 p T^{6} + 4656 p^{2} T^{7} + 372 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 8 T + 217 T^{2} + 2391 T^{3} + 29774 T^{4} + 260183 T^{5} + 29774 p T^{6} + 2391 p^{2} T^{7} + 217 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 7 T + 250 T^{2} - 903 T^{3} + 23765 T^{4} - 48253 T^{5} + 23765 p T^{6} - 903 p^{2} T^{7} + 250 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 21 T + 586 T^{2} - 7939 T^{3} + 121975 T^{4} - 1143691 T^{5} + 121975 p T^{6} - 7939 p^{2} T^{7} + 586 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.18181753617336024621698011764, −5.14002367959133247412397782537, −5.10339136200565389193831863709, −5.06928837648018263087341356769, −4.66580842458492710850646324819, −4.64391974113072798603675492978, −4.36850659751532971765998604641, −4.27727981340776338807403344506, −4.12208264666769934794363095592, −3.79887494849885598640095135593, −3.62495949893557746657739443652, −3.54385060295725947639879023941, −3.28912764668347652667513150028, −3.14674602602346814070299842064, −2.99503999804093788544092266048, −2.72399455022574741472333559072, −2.70082654331296460105221121326, −2.61400111471674258097005765193, −2.34263378196145346687608772081, −2.17929196518869179665182134975, −1.85094081854214560250877149282, −1.75021700119151168519816621475, −1.53625436142024455365372786184, −1.22860236796060971625632474926, −1.10975260985523699320964583679, 0, 0, 0, 0, 0, 1.10975260985523699320964583679, 1.22860236796060971625632474926, 1.53625436142024455365372786184, 1.75021700119151168519816621475, 1.85094081854214560250877149282, 2.17929196518869179665182134975, 2.34263378196145346687608772081, 2.61400111471674258097005765193, 2.70082654331296460105221121326, 2.72399455022574741472333559072, 2.99503999804093788544092266048, 3.14674602602346814070299842064, 3.28912764668347652667513150028, 3.54385060295725947639879023941, 3.62495949893557746657739443652, 3.79887494849885598640095135593, 4.12208264666769934794363095592, 4.27727981340776338807403344506, 4.36850659751532971765998604641, 4.64391974113072798603675492978, 4.66580842458492710850646324819, 5.06928837648018263087341356769, 5.10339136200565389193831863709, 5.14002367959133247412397782537, 5.18181753617336024621698011764

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