Properties

Label 2-2015-2015.1394-c0-0-1
Degree $2$
Conductor $2015$
Sign $0.702 - 0.711i$
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  + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + (0.866 − 1.5i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + (0.866 + 1.5i)23-s + 25-s − 31-s + (0.499 + 0.866i)36-s + (0.866 + 1.5i)37-s + (0.5 + 0.866i)41-s + (−0.866 + 1.5i)43-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)9-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + (0.866 − 1.5i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + (0.866 + 1.5i)23-s + 25-s − 31-s + (0.499 + 0.866i)36-s + (0.866 + 1.5i)37-s + (0.5 + 0.866i)41-s + (−0.866 + 1.5i)43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(1.00561\)
Root analytic conductor: \(1.00280\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ 0.702 - 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9081482446\)
\(L(\frac12)\) \(\approx\) \(0.9081482446\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.73T + T^{2} \)
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 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.73T + T^{2} \)
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

−9.388283705937069717362030820821, −8.591495608856959468633959486115, −7.87190168069418471209624532358, −7.25092327667428624919529815275, −6.52276719572584919219797994821, −5.24593808881362731225251243419, −4.36435858605497184682018966515, −3.57966518409483424019074569902, −3.11049624973495091304319654380, −1.14651985467179431511887067888, 0.863139815421126946163196074024, 2.20580174082125034035189673830, 3.69114515041586001311617957503, 4.30714991163580986441291342295, 5.19125041171068482694003410426, 5.96584377366271131378868977925, 6.94043380080173136519879875401, 7.75380354711148796049262955677, 8.603037814439052974861915891150, 8.975996713433514968821387944659

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