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Properties

Label 2-369600-1.1-c1-0-127
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s + 11-s + 2·13-s + 2·17-s − 4·19-s + 21-s − 4·23-s + 27-s − 6·29-s + 33-s − 6·37-s + 2·39-s + 10·41-s + 4·43-s + 49-s + 2·51-s − 2·53-s − 4·57-s − 4·59-s + 10·61-s + 63-s + 8·67-s − 4·69-s − 2·73-s + 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s + 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.986·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 0.529·57-s − 0.520·59-s + 1.28·61-s + 0.125·63-s + 0.977·67-s − 0.481·69-s − 0.234·73-s + 0.113·77-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: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.242513644\)
\(L(\frac12)\) \(\approx\) \(3.242513644\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−12.63231143448550, −12.14392050651807, −11.49569818304001, −11.14462683295945, −10.76851099749264, −10.14291043050380, −9.822136551678951, −9.265296502723060, −8.740092428964624, −8.542887302174546, −7.860009788179819, −7.610033613184850, −7.042200956856829, −6.496861148934620, −5.963235867131695, −5.589329623322149, −4.974118701854540, −4.301653018727691, −3.935120173594989, −3.571291664060047, −2.838021811560376, −2.265007611820221, −1.780106510257985, −1.224439464942377, −0.4461640620063619, 0.4461640620063619, 1.224439464942377, 1.780106510257985, 2.265007611820221, 2.838021811560376, 3.571291664060047, 3.935120173594989, 4.301653018727691, 4.974118701854540, 5.589329623322149, 5.963235867131695, 6.496861148934620, 7.042200956856829, 7.610033613184850, 7.860009788179819, 8.542887302174546, 8.740092428964624, 9.265296502723060, 9.822136551678951, 10.14291043050380, 10.76851099749264, 11.14462683295945, 11.49569818304001, 12.14392050651807, 12.63231143448550

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