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Properties

Label 4-3024e2-1.1-c1e2-0-10
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s + 6·11-s − 6·13-s + 4·17-s − 14·19-s + 23-s + 5·25-s + 2·29-s + 10·31-s − 35-s − 12·37-s − 8·41-s − 10·43-s − 8·47-s − 4·53-s − 6·55-s − 7·61-s + 6·65-s − 12·67-s + 30·71-s − 4·73-s + 6·77-s + 79-s − 12·83-s − 4·85-s − 8·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.80·11-s − 1.66·13-s + 0.970·17-s − 3.21·19-s + 0.208·23-s + 25-s + 0.371·29-s + 1.79·31-s − 0.169·35-s − 1.97·37-s − 1.24·41-s − 1.52·43-s − 1.16·47-s − 0.549·53-s − 0.809·55-s − 0.896·61-s + 0.744·65-s − 1.46·67-s + 3.56·71-s − 0.468·73-s + 0.683·77-s + 0.112·79-s − 1.31·83-s − 0.433·85-s − 0.847·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.094909305\)
\(L(\frac12)\) \(\approx\) \(1.094909305\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_z
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.19.o_dj
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2^2$ \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_az
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_cr
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_x
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_cf
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_r
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.61.h_am
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_cz
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.71.abe_od
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_ada
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.89.i_hm
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−9.057419668867284908765189274790, −8.394341078148482039632588681074, −8.368125679376854296902623333501, −7.899244469002196295053075009512, −7.34248227780917714588148214628, −6.84859327435795601663532680916, −6.76306777771102506681160583921, −6.29553034593679976182946176817, −6.18291423990719461187328971672, −5.24508511591339556192165483470, −4.95445013880922349234186907689, −4.54093334784865517096649220205, −4.46798244111652836455618468523, −3.62812930360360602220357648486, −3.51892137085955809714400187014, −2.84670739736882128859618692850, −2.28364808588270751632386977879, −1.73197498306471140245138132329, −1.34285373450897641264155794461, −0.32894135743839250840486763275, 0.32894135743839250840486763275, 1.34285373450897641264155794461, 1.73197498306471140245138132329, 2.28364808588270751632386977879, 2.84670739736882128859618692850, 3.51892137085955809714400187014, 3.62812930360360602220357648486, 4.46798244111652836455618468523, 4.54093334784865517096649220205, 4.95445013880922349234186907689, 5.24508511591339556192165483470, 6.18291423990719461187328971672, 6.29553034593679976182946176817, 6.76306777771102506681160583921, 6.84859327435795601663532680916, 7.34248227780917714588148214628, 7.899244469002196295053075009512, 8.368125679376854296902623333501, 8.394341078148482039632588681074, 9.057419668867284908765189274790

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