Properties

Label 2-369600-1.1-c1-0-138
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 − 3·17-s − 19-s − 21-s − 7·23-s + 27-s + 3·29-s + 4·31-s − 33-s + 8·37-s − 2·39-s + 6·41-s + 7·43-s + 10·47-s + 49-s − 3·51-s − 3·53-s − 57-s + 9·59-s − 9·61-s − 63-s − 12·67-s − 7·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.727·17-s − 0.229·19-s − 0.218·21-s − 1.45·23-s + 0.192·27-s + 0.557·29-s + 0.718·31-s − 0.174·33-s + 1.31·37-s − 0.320·39-s + 0.937·41-s + 1.06·43-s + 1.45·47-s + 1/7·49-s − 0.420·51-s − 0.412·53-s − 0.132·57-s + 1.17·59-s − 1.15·61-s − 0.125·63-s − 1.46·67-s − 0.842·69-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: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.768936729\)
\(L(\frac12)\) \(\approx\) \(2.768936729\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−12.50866831685527, −12.12272969205445, −11.72801877163942, −10.97415553724217, −10.76051664150381, −10.08337044349851, −9.821412838703098, −9.355946871656299, −8.837781422743307, −8.453190966304024, −7.866163469723923, −7.530709056425783, −7.104783899826098, −6.424150791019623, −5.989172012548128, −5.718329572009714, −4.677559981564797, −4.553390695242179, −4.006969365911414, −3.408940009250916, −2.751380974540420, −2.341659403335786, −2.000415955517083, −0.9931914633863256, −0.4583056883235691, 0.4583056883235691, 0.9931914633863256, 2.000415955517083, 2.341659403335786, 2.751380974540420, 3.408940009250916, 4.006969365911414, 4.553390695242179, 4.677559981564797, 5.718329572009714, 5.989172012548128, 6.424150791019623, 7.104783899826098, 7.530709056425783, 7.866163469723923, 8.453190966304024, 8.837781422743307, 9.355946871656299, 9.821412838703098, 10.08337044349851, 10.76051664150381, 10.97415553724217, 11.72801877163942, 12.12272969205445, 12.50866831685527

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