Properties

Label 2-369600-1.1-c1-0-14
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 + 5·13-s − 5·17-s − 4·19-s + 21-s − 7·23-s − 27-s + 3·29-s − 31-s + 33-s + 8·37-s − 5·39-s − 11·41-s − 43-s − 2·47-s + 49-s + 5·51-s + 3·53-s + 4·57-s − 11·59-s − 7·61-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 + 1.38·13-s − 1.21·17-s − 0.917·19-s + 0.218·21-s − 1.45·23-s − 0.192·27-s + 0.557·29-s − 0.179·31-s + 0.174·33-s + 1.31·37-s − 0.800·39-s − 1.71·41-s − 0.152·43-s − 0.291·47-s + 1/7·49-s + 0.700·51-s + 0.412·53-s + 0.529·57-s − 1.43·59-s − 0.896·61-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

\(L(1)\) \(\approx\) \(0.5039822248\)
\(L(\frac12)\) \(\approx\) \(0.5039822248\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−12.53234276095881, −11.98067241347230, −11.61863240279095, −11.05207064369183, −10.75951467262027, −10.37837197067318, −9.823021234441239, −9.419334246061864, −8.680486720230005, −8.530510424333404, −7.979229351680153, −7.444010149741775, −6.734031366965739, −6.453865447539669, −6.013785450198262, −5.751438619194852, −4.877051627033289, −4.506254794650317, −4.051866159365405, −3.496900037177142, −2.941030032592586, −2.150816720361691, −1.783497390377013, −1.035542057480887, −0.2051082377176234, 0.2051082377176234, 1.035542057480887, 1.783497390377013, 2.150816720361691, 2.941030032592586, 3.496900037177142, 4.051866159365405, 4.506254794650317, 4.877051627033289, 5.751438619194852, 6.013785450198262, 6.453865447539669, 6.734031366965739, 7.444010149741775, 7.979229351680153, 8.530510424333404, 8.680486720230005, 9.419334246061864, 9.823021234441239, 10.37837197067318, 10.75951467262027, 11.05207064369183, 11.61863240279095, 11.98067241347230, 12.53234276095881

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