Properties

Label 2-5328-37.36-c1-0-93
Degree $2$
Conductor $5328$
Sign $-0.367 - 0.929i$
Analytic cond. $42.5442$
Root an. cond. $6.52259$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.70i·5-s − 3.23·7-s − 6.47·11-s − 1.74i·13-s − 6.53i·17-s − 4.57i·19-s − 7.19i·23-s − 8.70·25-s − 4.78i·29-s − 6.32i·31-s + 11.9i·35-s + (2.23 + 5.65i)37-s − 2·41-s + 2.82i·43-s + 4·47-s + ⋯
L(s)  = 1  − 1.65i·5-s − 1.22·7-s − 1.95·11-s − 0.484i·13-s − 1.58i·17-s − 1.04i·19-s − 1.50i·23-s − 1.74·25-s − 0.888i·29-s − 1.13i·31-s + 2.02i·35-s + (0.367 + 0.929i)37-s − 0.312·41-s + 0.431i·43-s + 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $-0.367 - 0.929i$
Analytic conductor: \(42.5442\)
Root analytic conductor: \(6.52259\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (2737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5328,\ (\ :1/2),\ -0.367 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7139178519\)
\(L(\frac12)\) \(\approx\) \(0.7139178519\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37 \( 1 + (-2.23 - 5.65i)T \)
good5 \( 1 + 3.70iT - 5T^{2} \)
7 \( 1 + 3.23T + 7T^{2} \)
11 \( 1 + 6.47T + 11T^{2} \)
13 \( 1 + 1.74iT - 13T^{2} \)
17 \( 1 + 6.53iT - 17T^{2} \)
19 \( 1 + 4.57iT - 19T^{2} \)
23 \( 1 + 7.19iT - 23T^{2} \)
29 \( 1 + 4.78iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 2.82iT - 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 6.94T + 53T^{2} \)
59 \( 1 + 0.874iT - 59T^{2} \)
61 \( 1 + 5.65iT - 61T^{2} \)
67 \( 1 + 5.70T + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + 1.23T + 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + 7.19iT - 89T^{2} \)
97 \( 1 - 5.24iT - 97T^{2} \)
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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

−7.907754563480037481789792445933, −7.02468464940882132995007651707, −6.15461083249035167845526375320, −5.29051843108859394428296390903, −4.93305926251649896309844742157, −4.15684654291122258177209635369, −2.81336229131663283843331270311, −2.49669295902398102215397972322, −0.63217005681601753614186493473, −0.29267882587836521810988225107, 1.82848034087762946901225143695, 2.70659819205232415011118653218, 3.39780964378279882540944184186, 3.86273058746041351767313304672, 5.33659266976583365977852524031, 5.89764731188079222719167987653, 6.54099320279258945560355519923, 7.28515897737870458841867902345, 7.71365149327363461572092096839, 8.595156949645663589933500979017

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