Properties

Label 4-4864e2-1.1-c1e2-0-1
Degree $4$
Conductor $23658496$
Sign $1$
Analytic cond. $1508.48$
Root an. cond. $6.23211$
Motivic weight $1$
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  + 4·3-s + 6·9-s − 10·11-s + 10·17-s − 2·19-s + 25-s − 4·27-s − 40·33-s + 12·41-s + 2·43-s − 3·49-s + 40·51-s − 8·57-s + 12·59-s + 16·67-s + 18·73-s + 4·75-s − 37·81-s − 8·83-s + 8·89-s − 24·97-s − 60·99-s + 12·107-s + 36·113-s + 53·121-s + 48·123-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 2·9-s − 3.01·11-s + 2.42·17-s − 0.458·19-s + 1/5·25-s − 0.769·27-s − 6.96·33-s + 1.87·41-s + 0.304·43-s − 3/7·49-s + 5.60·51-s − 1.05·57-s + 1.56·59-s + 1.95·67-s + 2.10·73-s + 0.461·75-s − 4.11·81-s − 0.878·83-s + 0.847·89-s − 2.43·97-s − 6.03·99-s + 1.16·107-s + 3.38·113-s + 4.81·121-s + 4.32·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23658496\)    =    \(2^{16} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1508.48\)
Root analytic conductor: \(6.23211\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23658496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.477886601\)
\(L(\frac12)\) \(\approx\) \(4.477886601\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−8.199012514169048010737168226756, −8.131627992596824945468230516855, −7.87930343579529790731826243866, −7.62367063212017629532932322285, −7.31855882560195778873812215439, −6.96548661574832711600926573990, −6.05238573055749646551420815496, −5.96593777205801392874537438653, −5.41126341524580744494476640365, −5.11804763129789981238224074309, −4.93874498941538531093532671975, −4.05759765326545220143445358522, −3.74944005304287442355003589359, −3.43286410093743009506724981524, −2.92463138089217333325510541599, −2.68297459980156877217003186739, −2.35522274627593311533520259588, −2.10322838405441840472361406450, −1.21428956545367151980649732807, −0.48117586178068717928980774477, 0.48117586178068717928980774477, 1.21428956545367151980649732807, 2.10322838405441840472361406450, 2.35522274627593311533520259588, 2.68297459980156877217003186739, 2.92463138089217333325510541599, 3.43286410093743009506724981524, 3.74944005304287442355003589359, 4.05759765326545220143445358522, 4.93874498941538531093532671975, 5.11804763129789981238224074309, 5.41126341524580744494476640365, 5.96593777205801392874537438653, 6.05238573055749646551420815496, 6.96548661574832711600926573990, 7.31855882560195778873812215439, 7.62367063212017629532932322285, 7.87930343579529790731826243866, 8.131627992596824945468230516855, 8.199012514169048010737168226756

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