Properties

Label 4-304e2-1.1-c7e2-0-2
Degree $4$
Conductor $92416$
Sign $1$
Analytic cond. $9018.36$
Root an. cond. $9.74500$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 61·3-s + 175·5-s + 2.59e3·7-s − 899·9-s − 1.04e3·11-s + 1.46e4·13-s + 1.06e4·15-s + 2.96e4·17-s + 1.37e4·19-s + 1.58e5·21-s − 6.99e4·23-s + 2.06e4·25-s − 2.03e5·27-s + 1.38e5·29-s + 1.99e5·31-s − 6.37e4·33-s + 4.53e5·35-s − 6.78e4·37-s + 8.93e5·39-s − 5.39e5·41-s − 6.02e5·43-s − 1.57e5·45-s + 1.03e6·47-s + 3.41e6·49-s + 1.80e6·51-s + 2.13e6·53-s − 1.82e5·55-s + ⋯
L(s)  = 1  + 1.30·3-s + 0.626·5-s + 2.85·7-s − 0.411·9-s − 0.236·11-s + 1.84·13-s + 0.816·15-s + 1.46·17-s + 0.458·19-s + 3.72·21-s − 1.19·23-s + 0.264·25-s − 1.98·27-s + 1.05·29-s + 1.20·31-s − 0.308·33-s + 1.78·35-s − 0.220·37-s + 2.41·39-s − 1.22·41-s − 1.15·43-s − 0.257·45-s + 1.44·47-s + 4.14·49-s + 1.90·51-s + 1.97·53-s − 0.148·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(9018.36\)
Root analytic conductor: \(9.74500\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92416,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(14.67633918\)
\(L(\frac12)\) \(\approx\) \(14.67633918\)
\(L(\frac{9}{2})\) 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)$
bad2 \( 1 \)
19$C_1$ \( ( 1 - p^{3} T )^{2} \)
good3$D_{4}$ \( 1 - 61 T + 1540 p T^{2} - 61 p^{7} T^{3} + p^{14} T^{4} \)
5$D_{4}$ \( 1 - 7 p^{2} T + 398 p^{2} T^{2} - 7 p^{9} T^{3} + p^{14} T^{4} \)
7$D_{4}$ \( 1 - 2592 T + 3302069 T^{2} - 2592 p^{7} T^{3} + p^{14} T^{4} \)
11$D_{4}$ \( 1 + 95 p T - 11749120 T^{2} + 95 p^{8} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 - 14647 T + 173951598 T^{2} - 14647 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 - 29616 T + 1034411785 T^{2} - 29616 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 + 69985 T + 6725368502 T^{2} + 69985 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 - 138821 T + 15732402524 T^{2} - 138821 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 199396 T + 48643192158 T^{2} - 199396 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 + 67840 T + 47468074998 T^{2} + 67840 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 + 539350 T + 389443599074 T^{2} + 539350 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 + 602639 T + 634135307466 T^{2} + 602639 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 - 1031975 T + 749722035926 T^{2} - 1031975 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 - 2138263 T + 2496539389382 T^{2} - 2138263 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 - 3936369 T + 8841477061504 T^{2} - 3936369 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 1027655 T + 2220629234304 T^{2} + 1027655 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 + 764949 T + 9626156199098 T^{2} + 764949 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 3572084 T + 9205034831954 T^{2} - 3572084 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 - 9069522 T + 42655939394627 T^{2} - 9069522 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 2753414 T - 12495532808130 T^{2} - 2753414 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 - 7643046 T + 68606421453310 T^{2} - 7643046 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 1393620 T + 87948683189458 T^{2} - 1393620 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 + 6921466 T + 112774027874802 T^{2} + 6921466 p^{7} T^{3} + p^{14} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.62368621870460338874857636788, −10.38220992883789800798471665698, −9.769622342738715913038401964986, −9.121451844958210480653768739655, −8.510573482797269088336254734758, −8.359796558359768089906802549059, −8.032791194447450365453679712526, −7.927514306977531479563891486321, −6.99187385606731313786704923131, −6.29011908175732848067738947775, −5.53262493691892293627640459651, −5.39370590489020529088830037217, −4.85483537247859545618292944859, −3.92804001933779426909674858929, −3.64907771697253355902214683821, −2.87193467703827593812134311225, −2.12836878845878361751360473503, −1.95760919634578535083205091515, −1.02840949250476564253280079350, −0.943100352426913309290911331547, 0.943100352426913309290911331547, 1.02840949250476564253280079350, 1.95760919634578535083205091515, 2.12836878845878361751360473503, 2.87193467703827593812134311225, 3.64907771697253355902214683821, 3.92804001933779426909674858929, 4.85483537247859545618292944859, 5.39370590489020529088830037217, 5.53262493691892293627640459651, 6.29011908175732848067738947775, 6.99187385606731313786704923131, 7.927514306977531479563891486321, 8.032791194447450365453679712526, 8.359796558359768089906802549059, 8.510573482797269088336254734758, 9.121451844958210480653768739655, 9.769622342738715913038401964986, 10.38220992883789800798471665698, 10.62368621870460338874857636788

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