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Properties

Label 4-756e2-1.1-c1e2-0-12
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 3·7-s − 3·8-s + 6·11-s + 3·14-s − 16-s + 6·22-s + 6·23-s + 25-s − 3·28-s − 14·29-s + 5·32-s − 3·37-s + 12·43-s − 6·44-s + 6·46-s + 2·49-s + 50-s − 4·53-s − 9·56-s − 14·58-s + 7·64-s + 18·67-s + 18·71-s − 3·74-s + 18·77-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.13·7-s − 1.06·8-s + 1.80·11-s + 0.801·14-s − 1/4·16-s + 1.27·22-s + 1.25·23-s + 1/5·25-s − 0.566·28-s − 2.59·29-s + 0.883·32-s − 0.493·37-s + 1.82·43-s − 0.904·44-s + 0.884·46-s + 2/7·49-s + 0.141·50-s − 0.549·53-s − 1.20·56-s − 1.83·58-s + 7/8·64-s + 2.19·67-s + 2.13·71-s − 0.348·74-s + 2.05·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.951264786\)
\(L(\frac12)\) \(\approx\) \(2.951264786\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - T + p T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 - 3 T + p T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.11.ag_be
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.17.a_l
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.19.a_ah
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.23.ag_cc
29$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.o_du
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.d_ce
41$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.41.a_abl
43$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.43.am_ej
47$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.47.a_ax
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.59.a_b
61$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \) 2.61.a_adr
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.as_hr
71$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.71.as_io
73$C_2^2$ \( 1 - 47 T^{2} + p^{2} T^{4} \) 2.73.a_abv
79$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.j_gk
83$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \) 2.83.a_bx
89$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.89.a_eg
97$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \) 2.97.a_ach
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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

−8.671065156626678213396604679940, −7.890769815024314484364076496554, −7.60315567959625013589468326246, −6.97582199624844229940230611322, −6.60693348183691392840291153032, −6.03611098677557645113136270928, −5.52665390281788376448251679177, −5.13751990113151333491233414929, −4.70237638237238683147435528076, −4.01701542393498577462121399125, −3.83612529868128181061984586605, −3.27522885368086518119167085010, −2.35482163319595042775237315648, −1.64665366174566663065654309445, −0.860925924516051555995362908270, 0.860925924516051555995362908270, 1.64665366174566663065654309445, 2.35482163319595042775237315648, 3.27522885368086518119167085010, 3.83612529868128181061984586605, 4.01701542393498577462121399125, 4.70237638237238683147435528076, 5.13751990113151333491233414929, 5.52665390281788376448251679177, 6.03611098677557645113136270928, 6.60693348183691392840291153032, 6.97582199624844229940230611322, 7.60315567959625013589468326246, 7.890769815024314484364076496554, 8.671065156626678213396604679940

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