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Properties

Label 2-3724-133.37-c0-0-0
Degree $2$
Conductor $3724$
Sign $-0.701 - 0.712i$
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  + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−1 + 1.73i)23-s − 43-s + (−0.499 − 0.866i)45-s + (−0.5 + 0.866i)47-s − 0.999·55-s + (−0.5 + 0.866i)61-s + (−0.5 − 0.866i)73-s + (−0.499 − 0.866i)81-s − 2·83-s + 0.999·85-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−1 + 1.73i)23-s − 43-s + (−0.499 − 0.866i)45-s + (−0.5 + 0.866i)47-s − 0.999·55-s + (−0.5 + 0.866i)61-s + (−0.5 − 0.866i)73-s + (−0.499 − 0.866i)81-s − 2·83-s + 0.999·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ -0.701 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7909298258\)
\(L(\frac12)\) \(\approx\) \(0.7909298258\)
\(L(1)\) 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 \( 1 \)
19 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−9.048616211462397993847955943489, −8.057432989909275422487080002864, −7.35778004714409328051748199250, −7.04136499346802271328214882555, −6.04244948035794181222370909854, −5.14319191255763356976977400963, −4.44912547678862069342612598912, −3.42604861988246881576847567162, −2.69553600530230036484334596130, −1.69620310924035666304499455531, 0.44657338964641443807477173670, 1.66573020750314684042702900610, 3.02763530166876607375185831068, 3.87516854703353057433343640847, 4.42941452020313430966528628575, 5.52147903420193745335290785175, 6.20290676487627219566036145459, 6.76694191394180329297765370560, 8.030873348295406610039220135554, 8.484499916256035092218116157487

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