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Properties

Label 2-2015-2015.464-c0-0-7
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

\(L(\frac{1}{2})\) \(\approx\) \(1.646858013\)
\(L(\frac12)\) \(\approx\) \(1.646858013\)
\(L(1)\) 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

−8.973437830696771282244825353059, −8.433425281356616107840097949216, −7.38635798894064934213762770605, −6.67191979253310352585543787114, −6.07674539639486189808516648918, −5.36473189576526764261045781321, −3.94787282790723248587260014878, −2.73804189827363699147307946678, −1.82371456114366736857404340108, −1.18305123503357502726197551060, 2.33761764184055697666747542262, 3.03126444200120827247587212754, 3.83518184903051328584385722872, 4.66470846941597576815250329669, 5.27118353993412501623955610518, 6.33630492288886064595061644760, 7.73077641416160844032217067853, 8.446012274705330453190676880844, 8.835694282105224245604335065364, 9.564342273201083170771894671525

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