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Properties

Label 2-2015-2015.1394-c0-0-4
Degree $2$
Conductor $2015$
Sign $0.967 + 0.252i$
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  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + i·5-s + (0.866 + 0.499i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)10-s + (0.866 − 0.5i)11-s − 13-s − 0.999·14-s + (0.866 − 0.5i)15-s + (0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + (−0.499 + 0.866i)22-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + i·5-s + (0.866 + 0.499i)6-s + (0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)10-s + (0.866 − 0.5i)11-s − 13-s − 0.999·14-s + (0.866 − 0.5i)15-s + (0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + (−0.499 + 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $0.967 + 0.252i$
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: \((2,\ 2015,\ (\ :0),\ 0.967 + 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6332787423\)
\(L(\frac12)\) \(\approx\) \(0.6332787423\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 + T \)
31 \( 1 + T \)
good2 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.398121587405880735429065565557, −8.239501558596363181934064316133, −7.74970919740409789414442325945, −6.90604052070627467442963231600, −6.63187174909288567343098700000, −5.65856377549497123581631769828, −4.52533087607644128602241015050, −3.31238911242371124674191547865, −2.19352677406761870783467074308, −0.77814311817789044983980793330, 1.23353565337859570565436097616, 1.95717759675821167319041706783, 3.88102634312222432087052345024, 4.50490735826368934017673349916, 5.26587294270935180561486371593, 5.82566262596681075651698511869, 7.53128543714459648828370574123, 7.87043573792953607139541121471, 8.917246654485369860022971426350, 9.506832765195492912388797882326

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