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Properties

Label 2-2015-2015.464-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(2.036534073\)
\(L(\frac12)\) \(\approx\) \(2.036534073\)
\(L(1)\) 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

−9.251870867063070406454399412845, −8.565900037316766298991033388829, −7.30247933397961284366346906580, −6.93508351685398489942743926963, −6.35395400830920339656943634574, −5.64115311147651199572814929322, −4.46081993106333233479484009585, −3.49073516583723840525631366810, −2.78094736662485428386819332448, −1.50933507309928468738952462602, 1.35697326076553220875024877427, 2.98368547026787120932660667826, 3.75365517112442613391113443739, 4.05807006694459404990572403087, 4.94760572378684714890200114083, 5.90286168760770248318018196157, 6.75040124717050228992262232807, 8.058855341007834058645326891460, 8.793036275765040021888529734696, 9.222156917677372852900537656536

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