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Properties

Label 2-3696-924.587-c0-0-1
Degree $2$
Conductor $3696$
Sign $0.568 - 0.822i$
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  + (0.309 − 0.951i)3-s + (−0.951 + 0.690i)5-s + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (−0.587 + 0.809i)11-s + (0.363 + 1.11i)15-s + (−1.53 + 1.11i)17-s + (−0.190 + 0.587i)19-s − 0.999·21-s + 1.90·23-s + (0.118 − 0.363i)25-s + (−0.809 + 0.587i)27-s + (1.30 + 0.951i)31-s + (0.587 + 0.809i)33-s + (0.951 + 0.690i)35-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + (−0.951 + 0.690i)5-s + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)9-s + (−0.587 + 0.809i)11-s + (0.363 + 1.11i)15-s + (−1.53 + 1.11i)17-s + (−0.190 + 0.587i)19-s − 0.999·21-s + 1.90·23-s + (0.118 − 0.363i)25-s + (−0.809 + 0.587i)27-s + (1.30 + 0.951i)31-s + (0.587 + 0.809i)33-s + (0.951 + 0.690i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.568 - 0.822i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (3359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ 0.568 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7018834477\)
\(L(\frac12)\) \(\approx\) \(0.7018834477\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.587 - 0.809i)T \)
good5 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - 1.90T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + 1.90T + T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−8.505613724026449021654185353449, −8.012193479194270821041836762311, −7.18311306225019319706300812015, −6.87438766617267409874066798051, −6.25408069569263225164662764127, −4.89803184270733234097209765158, −4.08543087746508290465189391753, −3.26362072658380228995377665372, −2.48120760979730450651965443108, −1.25392933989046035436489334359, 0.40940740137555742144434832461, 2.60093707596343693606126814496, 2.90459069281215378944039648147, 4.13867699012286140413118694424, 4.72408671661436474790981215359, 5.33108237647092714787131284929, 6.24545928145029528355193448910, 7.25104439783959646824075227752, 8.185469104235857525068170763432, 8.672713939704851532490184448740

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