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Properties

Label 2-2015-2015.464-c0-0-6
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: $\chi_{2015} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ -0.967 + 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8724698265\)
\(L(\frac12)\) \(\approx\) \(0.8724698265\)
\(L(1)\) 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

−8.716391441165595649404890526151, −8.354232912556331504135807488136, −7.82637563437324004182774372122, −7.00854618579513910454885323749, −5.57543579479045702068846051341, −5.10952872245234090079413233872, −3.97654638410181062769900722985, −2.51203296321593883474084135356, −1.64454056673739804280620443546, −0.853013473635118497134698596767, 1.82662447873395131578870399265, 3.16579036118613786737294718931, 3.78192665371332889071938224063, 4.81284510964425801190112327088, 5.84650727057621951713300439644, 6.87278675372041595974301644151, 7.50516474929827856214133167838, 8.264161336744784686659322593765, 8.913919895241098777679019844245, 9.507353676290243259623646941164

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