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Properties

Label 2-3672-3672.3523-c0-0-0
Degree $2$
Conductor $3672$
Sign $-0.980 + 0.196i$
Analytic cond. $1.83256$
Root an. cond. $1.35372$
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  + (−0.984 + 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.766 − 0.642i)6-s + (−0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−1.80 − 0.157i)11-s + (0.866 + 0.499i)12-s + (0.766 − 0.642i)16-s + (−0.642 + 0.766i)17-s − i·18-s + (−1.32 + 0.766i)19-s + (1.80 − 0.157i)22-s + (−0.939 − 0.342i)24-s + (0.984 − 0.173i)25-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.642 + 0.766i)3-s + (0.939 − 0.342i)4-s + (−0.766 − 0.642i)6-s + (−0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−1.80 − 0.157i)11-s + (0.866 + 0.499i)12-s + (0.766 − 0.642i)16-s + (−0.642 + 0.766i)17-s − i·18-s + (−1.32 + 0.766i)19-s + (1.80 − 0.157i)22-s + (−0.939 − 0.342i)24-s + (0.984 − 0.173i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3672\)    =    \(2^{3} \cdot 3^{3} \cdot 17\)
Sign: $-0.980 + 0.196i$
Analytic conductor: \(1.83256\)
Root analytic conductor: \(1.35372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3672} (3523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3672,\ (\ :0),\ -0.980 + 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3479387681\)
\(L(\frac12)\) \(\approx\) \(0.3479387681\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (0.642 - 0.766i)T \)
good5 \( 1 + (-0.984 + 0.173i)T^{2} \)
7 \( 1 + (0.642 - 0.766i)T^{2} \)
11 \( 1 + (1.80 + 0.157i)T + (0.984 + 0.173i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
19 \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.642 - 0.766i)T^{2} \)
29 \( 1 + (-0.342 - 0.939i)T^{2} \)
31 \( 1 + (0.642 + 0.766i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.142 - 0.0999i)T + (0.342 - 0.939i)T^{2} \)
43 \( 1 + (1.26 + 1.50i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.826 + 0.984i)T + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.642 - 0.766i)T^{2} \)
67 \( 1 + (0.118 - 0.673i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (1.10 + 0.296i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.342 + 0.939i)T^{2} \)
83 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.157 - 1.80i)T + (-0.984 - 0.173i)T^{2} \)
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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

−8.739442705446972421648632608452, −8.566716355037630014188514780358, −7.936021536313338714478274618222, −7.15181436484722956785437279080, −6.19805863270373698322524635226, −5.39544931525559598001780212460, −4.60267972314135835790364241595, −3.48013425442555352248495294046, −2.58533026581897894526242958603, −1.89724307514953853660338432620, 0.22476316469203202821475754351, 1.70395669180545054670716388452, 2.66057474656153777755204380420, 2.97153424075395414552330619885, 4.42440537357480015662442663594, 5.46006313362090291111108444206, 6.55994217123571190036162670865, 6.99211285375804767067664714253, 7.72785052888382317926884743050, 8.408560144884896671033347130215

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