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Properties

Label 2-2015-2015.1239-c0-0-0
Degree $2$
Conductor $2015$
Sign $-0.962 + 0.271i$
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  + (−0.707 + 1.22i)3-s + (0.5 + 0.866i)4-s + i·5-s + (−0.499 − 0.866i)9-s − 1.41·12-s + (−0.965 − 0.258i)13-s + (−1.22 − 0.707i)15-s + (−0.499 + 0.866i)16-s + (−0.258 − 0.448i)17-s + (−0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + (0.258 − 0.448i)23-s − 25-s − i·31-s + (0.5 − 0.866i)36-s + (1.67 + 0.965i)37-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)3-s + (0.5 + 0.866i)4-s + i·5-s + (−0.499 − 0.866i)9-s − 1.41·12-s + (−0.965 − 0.258i)13-s + (−1.22 − 0.707i)15-s + (−0.499 + 0.866i)16-s + (−0.258 − 0.448i)17-s + (−0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + (0.258 − 0.448i)23-s − 25-s − i·31-s + (0.5 − 0.866i)36-s + (1.67 + 0.965i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.962 + 0.271i$
Analytic conductor: \(1.00561\)
Root analytic conductor: \(1.00280\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2015,\ (\ :0),\ -0.962 + 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7699952923\)
\(L(\frac12)\) \(\approx\) \(0.7699952923\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
31 \( 1 + iT \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.93T + T^{2} \)
59 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.517iT - T^{2} \)
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

−9.846873738844057689596061863259, −9.278398151920701907078100757614, −7.955807066878635686117839679986, −7.54670940188196031382830373201, −6.41213788295742204354607209833, −6.00085294561658541941478964753, −4.63224685801687549051381984427, −4.19914062305828782896550706239, −3.06238446534896681538535282432, −2.41187827335395227701244712834, 0.57446071008620327258759194942, 1.66822382485897813806817726776, 2.41197655677768046700255391762, 4.27965823257536400864728169352, 5.11896233763906585033289100549, 5.86853017789903756906387418641, 6.44714794099345335471252157667, 7.30317102368891381796007105369, 7.83601458231890239735638887189, 9.094648526852402433530552590629

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