Properties

Label 2-3696-924.251-c0-0-3
Degree $2$
Conductor $3696$
Sign $-0.998 + 0.0457i$
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  + (0.809 − 0.587i)3-s + (−0.587 − 1.80i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (−1.53 − 1.11i)15-s + (0.363 + 1.11i)17-s + (1.30 − 0.951i)19-s − 21-s − 1.17·23-s + (−2.11 + 1.53i)25-s + (−0.309 − 0.951i)27-s + (−0.190 + 0.587i)31-s + (−0.951 + 0.309i)33-s + (−0.587 + 1.80i)35-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.587 − 1.80i)5-s + (−0.809 − 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (−1.53 − 1.11i)15-s + (0.363 + 1.11i)17-s + (1.30 − 0.951i)19-s − 21-s − 1.17·23-s + (−2.11 + 1.53i)25-s + (−0.309 − 0.951i)27-s + (−0.190 + 0.587i)31-s + (−0.951 + 0.309i)33-s + (−0.587 + 1.80i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.998 + 0.0457i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (3023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ -0.998 + 0.0457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075126554\)
\(L(\frac12)\) \(\approx\) \(1.075126554\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.951 + 0.309i)T \)
good5 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.17T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + 1.17T + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−8.268061988298508502009292261088, −7.75768861475043386457022687972, −7.19566272909059935646616870099, −6.03583249497449403997268471201, −5.36101765207498669829490868611, −4.31014400734032095180165396701, −3.73465920113198479072491902616, −2.81096374143123245887085625999, −1.50061263828955920679368889993, −0.55210809854561069690562837197, 2.33081008907206911532924080379, 2.81314135606133321521601286145, 3.43243073441645266878428113684, 4.18368478002953983523449995656, 5.40692549224645970613982802396, 6.10126166023432475640452591824, 7.08539997163606391229571044186, 7.75090318884627669509567094069, 7.971766567316372533240957566924, 9.331550516843396145517753814225

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