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Properties

Label 2-369600-1.1-c1-0-131
Degree $2$
Conductor $369600$
Sign $-1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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  − 3-s + 7-s + 9-s − 11-s − 4·13-s − 3·17-s − 7·19-s − 21-s − 9·23-s − 27-s + 3·29-s − 2·31-s + 33-s − 4·37-s + 4·39-s − 6·41-s + 43-s − 6·47-s + 49-s + 3·51-s + 3·53-s + 7·57-s − 9·59-s + 61-s + 63-s + 10·67-s + 9·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.727·17-s − 1.60·19-s − 0.218·21-s − 1.87·23-s − 0.192·27-s + 0.557·29-s − 0.359·31-s + 0.174·33-s − 0.657·37-s + 0.640·39-s − 0.937·41-s + 0.152·43-s − 0.875·47-s + 1/7·49-s + 0.420·51-s + 0.412·53-s + 0.927·57-s − 1.17·59-s + 0.128·61-s + 0.125·63-s + 1.22·67-s + 1.08·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−12.57771600695674, −12.29831933925157, −11.84973894546283, −11.41370140039353, −10.88776887346286, −10.42877365981826, −10.19716496138282, −9.625473579251088, −9.177741754225218, −8.424513690415039, −8.236344586030116, −7.772625877233864, −7.020961447778212, −6.773675306691100, −6.300711874007179, −5.637063969500654, −5.358503495171677, −4.632189622703133, −4.338474523614843, −3.954998146170849, −3.075603304914974, −2.513846432032208, −1.836232224202730, −1.696555863581201, −0.4517698815565125, 0, 0.4517698815565125, 1.696555863581201, 1.836232224202730, 2.513846432032208, 3.075603304914974, 3.954998146170849, 4.338474523614843, 4.632189622703133, 5.358503495171677, 5.637063969500654, 6.300711874007179, 6.773675306691100, 7.020961447778212, 7.772625877233864, 8.236344586030116, 8.424513690415039, 9.177741754225218, 9.625473579251088, 10.19716496138282, 10.42877365981826, 10.88776887346286, 11.41370140039353, 11.84973894546283, 12.29831933925157, 12.57771600695674

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