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Properties

Label 2-3724-931.835-c0-0-0
Degree $2$
Conductor $3724$
Sign $0.325 - 0.945i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.147 + 0.0222i)5-s + (0.623 + 0.781i)7-s + (−0.733 − 0.680i)9-s + (0.326 − 0.302i)11-s + (0.123 + 1.64i)17-s + (−0.5 + 0.866i)19-s + (−0.109 + 1.46i)23-s + (−0.934 + 0.288i)25-s + (−0.109 − 0.101i)35-s + (1.03 − 1.29i)43-s + (0.123 + 0.0841i)45-s + (1.19 + 0.367i)47-s + (−0.222 + 0.974i)49-s + (−0.0414 + 0.0520i)55-s + (−0.367 + 0.250i)61-s + ⋯
L(s)  = 1  + (−0.147 + 0.0222i)5-s + (0.623 + 0.781i)7-s + (−0.733 − 0.680i)9-s + (0.326 − 0.302i)11-s + (0.123 + 1.64i)17-s + (−0.5 + 0.866i)19-s + (−0.109 + 1.46i)23-s + (−0.934 + 0.288i)25-s + (−0.109 − 0.101i)35-s + (1.03 − 1.29i)43-s + (0.123 + 0.0841i)45-s + (1.19 + 0.367i)47-s + (−0.222 + 0.974i)49-s + (−0.0414 + 0.0520i)55-s + (−0.367 + 0.250i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (2697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.325 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126062518\)
\(L(\frac12)\) \(\approx\) \(1.126062518\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.733 + 0.680i)T^{2} \)
5 \( 1 + (0.147 - 0.0222i)T + (0.955 - 0.294i)T^{2} \)
11 \( 1 + (-0.326 + 0.302i)T + (0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.123 - 1.64i)T + (-0.988 + 0.149i)T^{2} \)
23 \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-1.03 + 1.29i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-1.19 - 0.367i)T + (0.826 + 0.563i)T^{2} \)
53 \( 1 + (-0.365 - 0.930i)T^{2} \)
59 \( 1 + (-0.955 - 0.294i)T^{2} \)
61 \( 1 + (0.367 - 0.250i)T + (0.365 - 0.930i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.91 + 0.589i)T + (0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.0747 - 0.997i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.861298174463837178002100996686, −8.107849811791018874723571056489, −7.61771245445280761989544830818, −6.36792598577234351728054798713, −5.85742446949397370730030462208, −5.34794458429710145997514328862, −3.96296262719699062837096441619, −3.61925532750308278196822847884, −2.34199442303453918256386676525, −1.43801208066120929083031849009, 0.66259167544785012370933669529, 2.15638512218014445912180281272, 2.86597756072453365318302850623, 4.16618188695783181813776576120, 4.66519150255053058755897358483, 5.41867639342924537154388930609, 6.44171415068684720625661390562, 7.17512544098289330007158446015, 7.80845883817131689070398440599, 8.459239684384883727223143686902

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