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Properties

Label 2-2070-15.8-c1-0-14
Degree $2$
Conductor $2070$
Sign $0.794 - 0.607i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
Motivic weight $1$
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 − 0.707i)2-s + 1.00i·4-s + (1.85 − 1.24i)5-s + (−2 + 2i)7-s + (0.707 − 0.707i)8-s + (−2.19 − 0.432i)10-s + 5.31i·11-s + (−0.864 − 0.864i)13-s + 2.82·14-s − 1.00·16-s + (−1.41 − 1.41i)17-s − 6.38i·19-s + (1.24 + 1.85i)20-s + (3.76 − 3.76i)22-s + (0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.830 − 0.557i)5-s + (−0.755 + 0.755i)7-s + (0.250 − 0.250i)8-s + (−0.693 − 0.136i)10-s + 1.60i·11-s + (−0.239 − 0.239i)13-s + 0.755·14-s − 0.250·16-s + (−0.342 − 0.342i)17-s − 1.46i·19-s + (0.278 + 0.415i)20-s + (0.801 − 0.801i)22-s + (0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.201585977\)
\(L(\frac12)\) \(\approx\) \(1.201585977\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-1.85 + 1.24i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 - 5.31iT - 11T^{2} \)
13 \( 1 + (0.864 + 0.864i)T + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 + 6.38iT - 19T^{2} \)
29 \( 1 - 4.05T + 29T^{2} \)
31 \( 1 - 5.52T + 31T^{2} \)
37 \( 1 + (5.76 - 5.76i)T - 37iT^{2} \)
41 \( 1 - 11.6iT - 41T^{2} \)
43 \( 1 + (-9.01 - 9.01i)T + 43iT^{2} \)
47 \( 1 + (-0.885 - 0.885i)T + 47iT^{2} \)
53 \( 1 + (6.81 - 6.81i)T - 53iT^{2} \)
59 \( 1 + 2.15T + 59T^{2} \)
61 \( 1 - 7.91T + 61T^{2} \)
67 \( 1 + (-4.35 + 4.35i)T - 67iT^{2} \)
71 \( 1 - 8.00iT - 71T^{2} \)
73 \( 1 + (8.25 + 8.25i)T + 73iT^{2} \)
79 \( 1 - 8.77iT - 79T^{2} \)
83 \( 1 + (2.07 - 2.07i)T - 83iT^{2} \)
89 \( 1 - 9.32T + 89T^{2} \)
97 \( 1 + (6.38 - 6.38i)T - 97iT^{2} \)
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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

−9.404212236917114560842767231308, −8.729340544235784151866981745094, −7.78793163239912047078886231882, −6.76988567980944043838249290784, −6.24520111559381273757644298022, −4.89971960175198600954086700448, −4.58753834395722490845903870908, −2.82855557039782722199272762341, −2.44293187728820223419248192421, −1.14842543064496454712299257126, 0.55641832366056756113095562266, 1.95013616213426160655675450322, 3.20157647319302319259785189122, 3.97491994755053798655430314398, 5.48060592868907635498130309192, 5.97493152544170184531746545222, 6.71081285319222504812090005236, 7.33323556574941076426839348380, 8.379533746508188119979967062120, 8.959137290904482656938101868261

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