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Properties

Label 2-2015-2015.464-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.7958546839\)
\(L(\frac12)\) \(\approx\) \(0.7958546839\)
\(L(1)\) 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

−9.696458360559740036082598044644, −9.507917819579864121562677269306, −7.80506198302458931224383098257, −7.35155686067828106775903591406, −6.16941293966558431481650979897, −5.69042756035483501384586770668, −5.26118845461006862789304720285, −3.94644261791557674409029061517, −3.54373597283127229743767133190, −2.33943313506733750746649526193, 0.42879962757382995677236944990, 1.99356432997582182125251116878, 2.98170996496077595740153745287, 4.03457583952622186415759466291, 4.96059131928207351612373202086, 5.39652257485490688831361732314, 6.48478576315726184687172186742, 7.44295490104108815472833467249, 7.78393121458017243644023875441, 9.075208542700394679648943933454

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