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Properties

Label 2-2015-2015.464-c0-0-3
Degree $2$
Conductor $2015$
Sign $0.711 - 0.702i$
Analytic cond. $1.00561$
Root an. cond. $1.00280$
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  + (−1 + 1.73i)3-s + (−0.5 − 0.866i)4-s + 5-s + (−1.49 − 2.59i)9-s + 1.99·12-s + (−0.5 − 0.866i)13-s + (−1 + 1.73i)15-s + (−0.499 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + (0.5 − 0.866i)23-s + 25-s + 4·27-s + 31-s + ⋯
L(s)  = 1  + (−1 + 1.73i)3-s + (−0.5 − 0.866i)4-s + 5-s + (−1.49 − 2.59i)9-s + 1.99·12-s + (−0.5 − 0.866i)13-s + (−1 + 1.73i)15-s + (−0.499 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)20-s + (0.5 − 0.866i)23-s + 25-s + 4·27-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(1.00561\)
Root analytic conductor: \(1.00280\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ 0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8460622038\)
\(L(\frac12)\) \(\approx\) \(0.8460622038\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T + T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.676642283123131507476408982095, −9.129322024308572396869843617420, −8.205981176130642541653464924154, −6.54142234799390534556624658862, −5.86640189317603337710717035399, −5.48659295241841851870137711904, −4.75986483817534328164296509563, −3.98900634299216580074548649290, −2.82478551786765405845661030346, −0.986569720624933608347739662660, 0.990568801415586614304496020758, 2.21619469952025777608845809487, 2.98483254374812149878381685788, 4.88976290515679562469307580999, 5.16078195795805571652868798366, 6.32369784988611678914749843142, 6.87961465135138926115936289097, 7.51049586488008425549283797497, 8.224537453385269541028352786251, 9.201168240018315373142171739280

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