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Properties

Label 4-676000-1.1-c1e2-0-17
Degree $4$
Conductor $676000$
Sign $1$
Analytic cond. $43.1023$
Root an. cond. $2.56227$
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 − 5-s + 8-s − 9-s − 10-s + 5·13-s + 16-s + 9·17-s − 18-s − 20-s + 25-s + 5·26-s + 15·29-s + 32-s + 9·34-s − 36-s + 14·37-s − 40-s + 6·41-s + 45-s − 4·49-s + 50-s + 5·52-s − 18·53-s + 15·58-s − 13·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 1/3·9-s − 0.316·10-s + 1.38·13-s + 1/4·16-s + 2.18·17-s − 0.235·18-s − 0.223·20-s + 1/5·25-s + 0.980·26-s + 2.78·29-s + 0.176·32-s + 1.54·34-s − 1/6·36-s + 2.30·37-s − 0.158·40-s + 0.937·41-s + 0.149·45-s − 4/7·49-s + 0.141·50-s + 0.693·52-s − 2.47·53-s + 1.96·58-s − 1.66·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676000\)    =    \(2^{5} \cdot 5^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(43.1023\)
Root analytic conductor: \(2.56227\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.630714895\)
\(L(\frac12)\) \(\approx\) \(3.630714895\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$ \( 1 + T \)
13$C_2$ \( 1 - 5 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.11.a_i
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.17.aj_ca
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.19.a_aq
23$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.23.a_u
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.ap_ei
31$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.31.a_f
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.37.ao_ek
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ag_de
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.43.a_bm
47$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.47.a_e
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.59.a_abc
61$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.n_gg
67$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \) 2.67.a_cj
71$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.71.a_l
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.n_gm
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2^2$ \( 1 - 137 T^{2} + p^{2} T^{4} \) 2.83.a_afh
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.m_gw
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.i_gs
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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

−8.175291204750518296320087128236, −7.902173944596275852983072295458, −7.57917758355389146139677727602, −6.91970508698476870540250485373, −6.31745033743705419764734728886, −6.01163889640874395930141462502, −5.79984888153536237599334897760, −5.02321237334432265542234888930, −4.47144635926184265951641904059, −4.26462123498517851377218225377, −3.34053688445705834503141420641, −3.12675343315169675519174726281, −2.68193217175934625591830962311, −1.42875223154882824758743397791, −0.983659185130878969177250493778, 0.983659185130878969177250493778, 1.42875223154882824758743397791, 2.68193217175934625591830962311, 3.12675343315169675519174726281, 3.34053688445705834503141420641, 4.26462123498517851377218225377, 4.47144635926184265951641904059, 5.02321237334432265542234888930, 5.79984888153536237599334897760, 6.01163889640874395930141462502, 6.31745033743705419764734728886, 6.91970508698476870540250485373, 7.57917758355389146139677727602, 7.902173944596275852983072295458, 8.175291204750518296320087128236

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