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Properties

Label 2-397800-1.1-c1-0-106
Degree $2$
Conductor $397800$
Sign $1$
Analytic cond. $3176.44$
Root an. cond. $56.3599$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 5·11-s − 13-s − 17-s − 3·19-s − 6·23-s + 5·29-s + 8·31-s − 7·37-s + 7·41-s − 9·47-s − 6·49-s − 11·53-s − 6·59-s − 8·61-s − 2·67-s + 12·71-s − 9·73-s − 5·77-s + 12·83-s − 16·89-s − 91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.50·11-s − 0.277·13-s − 0.242·17-s − 0.688·19-s − 1.25·23-s + 0.928·29-s + 1.43·31-s − 1.15·37-s + 1.09·41-s − 1.31·47-s − 6/7·49-s − 1.51·53-s − 0.781·59-s − 1.02·61-s − 0.244·67-s + 1.42·71-s − 1.05·73-s − 0.569·77-s + 1.31·83-s − 1.69·89-s − 0.104·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(397800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(3176.44\)
Root analytic conductor: \(56.3599\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 397800,\ (\ :1/2),\ 1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.76010078028740, −12.41920080856948, −12.13111048014744, −11.49563315161359, −11.00733232800910, −10.61974004387648, −10.29927802365370, −9.685005607716455, −9.487827895047938, −8.616297898847929, −8.205277185455000, −8.028081463104505, −7.578392944211367, −6.877987003796077, −6.406547746997376, −6.039150890062695, −5.395685950210937, −4.892790375883207, −4.560131764126809, −4.100623710009141, −3.234776655318041, −2.866065558060346, −2.312778904250368, −1.783797172259607, −1.136856902668814, 0, 0, 1.136856902668814, 1.783797172259607, 2.312778904250368, 2.866065558060346, 3.234776655318041, 4.100623710009141, 4.560131764126809, 4.892790375883207, 5.395685950210937, 6.039150890062695, 6.406547746997376, 6.877987003796077, 7.578392944211367, 8.028081463104505, 8.205277185455000, 8.616297898847929, 9.487827895047938, 9.685005607716455, 10.29927802365370, 10.61974004387648, 11.00733232800910, 11.49563315161359, 12.13111048014744, 12.41920080856948, 12.76010078028740

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