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Properties

Label 2-3672-1224.1091-c0-0-0
Degree $2$
Conductor $3672$
Sign $-0.0117 - 0.999i$
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  + (−0.991 − 0.130i)2-s + (0.965 + 0.258i)4-s + (−0.923 − 0.382i)8-s + (−0.608 + 1.79i)11-s + (0.866 + 0.5i)16-s + (0.130 − 0.991i)17-s + (−0.478 + 0.198i)19-s + (0.837 − 1.69i)22-s + (0.608 + 0.793i)25-s + (−0.793 − 0.608i)32-s + (−0.258 + 0.965i)34-s + (0.5 − 0.133i)38-s + (−0.0420 + 0.641i)41-s + (−0.207 + 0.158i)43-s + (−1.05 + 1.57i)44-s + ⋯
L(s)  = 1  + (−0.991 − 0.130i)2-s + (0.965 + 0.258i)4-s + (−0.923 − 0.382i)8-s + (−0.608 + 1.79i)11-s + (0.866 + 0.5i)16-s + (0.130 − 0.991i)17-s + (−0.478 + 0.198i)19-s + (0.837 − 1.69i)22-s + (0.608 + 0.793i)25-s + (−0.793 − 0.608i)32-s + (−0.258 + 0.965i)34-s + (0.5 − 0.133i)38-s + (−0.0420 + 0.641i)41-s + (−0.207 + 0.158i)43-s + (−1.05 + 1.57i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0117 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3672\)    =    \(2^{3} \cdot 3^{3} \cdot 17\)
Sign: $-0.0117 - 0.999i$
Analytic conductor: \(1.83256\)
Root analytic conductor: \(1.35372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3672} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3672,\ (\ :0),\ -0.0117 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5928513194\)
\(L(\frac12)\) \(\approx\) \(0.5928513194\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.991 + 0.130i)T \)
3 \( 1 \)
17 \( 1 + (-0.130 + 0.991i)T \)
good5 \( 1 + (-0.608 - 0.793i)T^{2} \)
7 \( 1 + (0.608 - 0.793i)T^{2} \)
11 \( 1 + (0.608 - 1.79i)T + (-0.793 - 0.608i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.478 - 0.198i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.130 + 0.991i)T^{2} \)
29 \( 1 + (-0.991 + 0.130i)T^{2} \)
31 \( 1 + (-0.793 + 0.608i)T^{2} \)
37 \( 1 + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (0.0420 - 0.641i)T + (-0.991 - 0.130i)T^{2} \)
43 \( 1 + (0.207 - 0.158i)T + (0.258 - 0.965i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.158 - 1.20i)T + (-0.965 + 0.258i)T^{2} \)
61 \( 1 + (-0.608 + 0.793i)T^{2} \)
67 \( 1 + (1.71 - 0.991i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.923 + 0.382i)T^{2} \)
73 \( 1 + (0.732 + 1.09i)T + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (-0.793 - 0.608i)T^{2} \)
83 \( 1 + (0.241 - 1.83i)T + (-0.965 - 0.258i)T^{2} \)
89 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
97 \( 1 + (-0.0578 - 0.882i)T + (-0.991 + 0.130i)T^{2} \)
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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

−9.039179928988584107856280322927, −8.131873654556211915546048386264, −7.38719490310577816803344821179, −7.07250107802853525725437113684, −6.12773386128178442684478525332, −5.11936385115569610950385196323, −4.35540663347724528528863407937, −3.09553550835382774388905601526, −2.33943166229386820925526321785, −1.37042009770264623608669407651, 0.47594463667007018661501477614, 1.79516725350215514788640528761, 2.87005435431963587082104517775, 3.58684666376833297812807661177, 4.90143841509416643813811604569, 5.91884504455420204717441719350, 6.22509648850229349519166756850, 7.18758348338288374480787280790, 8.038405547284462454801996236500, 8.556795390539968581176206275761

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