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Properties

Label 2-2015-2015.1239-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.656680041\)
\(L(\frac12)\) \(\approx\) \(1.656680041\)
\(L(1)\) 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

−8.977179148292978440706209865691, −8.403908955872186462880472872705, −7.67194041077615626498209065046, −7.19874613088871622234704929032, −6.46143699122784550105575090299, −5.86386526818525524446732779596, −3.95056662339358267455064680552, −3.45069881372926555137216839481, −2.36026941885050001909222793336, −1.80538041481446758376440297499, 1.21985207921428202584842022665, 2.51339167276350745531615468180, 3.57823066736154014722451361467, 4.51734204961240683035154922878, 5.09890343149857407930072208889, 5.96229441960981974182359803508, 6.86414538366176651209642451678, 8.089911324481658318774789555178, 8.781725949598687272514657021153, 9.229767790381110840260669870353

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