Properties

Label 4-2738e2-1.1-c1e2-0-6
Degree $4$
Conductor $7496644$
Sign $1$
Analytic cond. $477.992$
Root an. cond. $4.67579$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 3·4-s − 3·5-s + 2·6-s − 4·7-s + 4·8-s − 6·10-s − 3·11-s + 3·12-s − 3·13-s − 8·14-s − 3·15-s + 5·16-s + 6·17-s − 6·19-s − 9·20-s − 4·21-s − 6·22-s − 3·23-s + 4·24-s + 2·25-s − 6·26-s + 2·27-s − 12·28-s − 3·29-s − 6·30-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3/2·4-s − 1.34·5-s + 0.816·6-s − 1.51·7-s + 1.41·8-s − 1.89·10-s − 0.904·11-s + 0.866·12-s − 0.832·13-s − 2.13·14-s − 0.774·15-s + 5/4·16-s + 1.45·17-s − 1.37·19-s − 2.01·20-s − 0.872·21-s − 1.27·22-s − 0.625·23-s + 0.816·24-s + 2/5·25-s − 1.17·26-s + 0.384·27-s − 2.26·28-s − 0.557·29-s − 1.09·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7496644\)    =    \(2^{2} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(477.992\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7496644,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
37 \( 1 \)
good3$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_b
5$D_{4}$ \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_h
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_t
13$D_{4}$ \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_x
17$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_w
19$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_ba
23$D_{4}$ \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_br
29$D_{4}$ \( 1 + 3 T + 55 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_cd
31$D_{4}$ \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.31.ad_r
41$D_{4}$ \( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.41.p_fd
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.43.m_es
47$D_{4}$ \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_de
53$D_{4}$ \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_dq
59$D_{4}$ \( 1 + 6 T + 106 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_ec
61$D_{4}$ \( 1 + 21 T + 227 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.61.v_it
67$D_{4}$ \( 1 - T + 87 T^{2} - p T^{3} + p^{2} T^{4} \) 2.67.ab_dj
71$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.71.a_cg
73$D_{4}$ \( 1 + 5 T + 105 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.73.f_eb
79$D_{4}$ \( 1 - 21 T + 263 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.79.av_kd
83$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_eo
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$D_{4}$ \( 1 + 18 T + 254 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.97.s_ju
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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

−8.430049700660278996550799263120, −8.231787891512325700239599211875, −7.72171211054302389873152461519, −7.50397370992441951624202857721, −7.14933880785220646855591311992, −6.55139286803122870310704217021, −6.34643379037890045416204545226, −6.08105898349458993540120145324, −5.36009470265819444175475468515, −4.95618354491442827153290424462, −4.78974976684770228697750357450, −4.16075339788162996450092463231, −3.58103824280679124920555302235, −3.55079034813787560044898625385, −2.96288475815850050195052766523, −2.86189672270611689661654739208, −2.07383749064450879319630605217, −1.52540210086806738473601900148, 0, 0, 1.52540210086806738473601900148, 2.07383749064450879319630605217, 2.86189672270611689661654739208, 2.96288475815850050195052766523, 3.55079034813787560044898625385, 3.58103824280679124920555302235, 4.16075339788162996450092463231, 4.78974976684770228697750357450, 4.95618354491442827153290424462, 5.36009470265819444175475468515, 6.08105898349458993540120145324, 6.34643379037890045416204545226, 6.55139286803122870310704217021, 7.14933880785220646855591311992, 7.50397370992441951624202857721, 7.72171211054302389873152461519, 8.231787891512325700239599211875, 8.430049700660278996550799263120

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