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Properties

Label 4-676000-1.1-c1e2-0-35
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s − 9-s + 10-s − 7·13-s + 16-s − 3·17-s − 18-s + 20-s + 25-s − 7·26-s + 3·29-s + 32-s − 3·34-s − 36-s + 2·37-s + 40-s − 6·41-s − 45-s + 8·49-s + 50-s − 7·52-s + 6·53-s + 3·58-s − 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 − 1.94·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.223·20-s + 1/5·25-s − 1.37·26-s + 0.557·29-s + 0.176·32-s − 0.514·34-s − 1/6·36-s + 0.328·37-s + 0.158·40-s − 0.937·41-s − 0.149·45-s + 8/7·49-s + 0.141·50-s − 0.970·52-s + 0.824·53-s + 0.393·58-s − 0.128·61-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: \(1\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
5$C_1$ \( 1 - T \)
13$C_2$ \( 1 + 7 T + p T^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.7.a_ai
11$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.11.a_aq
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.d_q
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.23.a_ae
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.ad_bo
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.a_at
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ac_co
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.47.a_bo
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.59.a_dc
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.b_co
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.a_n
71$C_2^2$ \( 1 + 107 T^{2} + p^{2} T^{4} \) 2.71.a_ed
73$C_2$$\times$$C_2$ \( ( 1 + 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.z_lo
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.83.a_h
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

−8.174400987184276985369058419126, −7.41996366081885346916609967041, −7.13333556365983389556273859356, −6.88075667321381553697297779880, −6.21421891582018773916112606458, −5.74052442482770183921988742681, −5.37224113674013722329590717298, −4.82569909063631226748549959828, −4.44293261634869723212399878383, −3.97147009700585062565317027139, −3.08378908252782656312046553908, −2.64252302487584719930389527367, −2.25786892163207388315159645385, −1.39135393562036753222838259329, 0, 1.39135393562036753222838259329, 2.25786892163207388315159645385, 2.64252302487584719930389527367, 3.08378908252782656312046553908, 3.97147009700585062565317027139, 4.44293261634869723212399878383, 4.82569909063631226748549959828, 5.37224113674013722329590717298, 5.74052442482770183921988742681, 6.21421891582018773916112606458, 6.88075667321381553697297779880, 7.13333556365983389556273859356, 7.41996366081885346916609967041, 8.174400987184276985369058419126

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