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Properties

Label 2-3696-924.263-c0-0-0
Degree $2$
Conductor $3696$
Sign $0.553 - 0.832i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
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)3-s + i·7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 + 1.5i)17-s + (0.866 + 1.5i)19-s + (0.866 + 0.5i)21-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 0.999·27-s − 1.73·29-s + (0.499 + 0.866i)33-s + (−0.5 − 0.866i)37-s + 1.73·43-s + (−0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + i·7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 + 1.5i)17-s + (0.866 + 1.5i)19-s + (0.866 + 0.5i)21-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 0.999·27-s − 1.73·29-s + (0.499 + 0.866i)33-s + (−0.5 − 0.866i)37-s + 1.73·43-s + (−0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (2111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ 0.553 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186880116\)
\(L(\frac12)\) \(\approx\) \(1.186880116\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.832909057153361463614361747198, −7.88329139819336452046363626873, −7.57840139180861195583709314051, −6.68258483380097638955441149752, −5.70688351694219780949466873411, −5.46106238750197611231552952308, −3.96362956022366299055582214573, −3.32241607880670991703698657744, −2.02571192057638305796016533362, −1.77476509633370357473832612849, 0.60597880839510910989723466885, 2.44931616214358870292476036837, 3.04463560323983282714909172280, 3.99817230547612418261496265252, 4.75014743788236206371124603048, 5.29856765855697138768793354175, 6.42759722695732628661561991159, 7.31163873713961861327449649389, 7.80014707911684214206552202814, 8.802246571558164112236348462623

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