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Properties

Label 2-1890-315.59-c1-0-26
Degree $2$
Conductor $1890$
Sign $0.966 + 0.255i$
Analytic cond. $15.0917$
Root an. cond. $3.88480$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.09 + 0.770i)5-s + (1.47 − 2.19i)7-s − 0.999·8-s + (1.71 − 1.43i)10-s + 3.61i·11-s + (−1.68 + 2.92i)13-s + (−1.16 − 2.37i)14-s + (−0.5 + 0.866i)16-s + (6.40 + 3.69i)17-s + (−4.15 + 2.39i)19-s + (−0.381 − 2.20i)20-s + (3.12 + 1.80i)22-s + 7.98·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.938 + 0.344i)5-s + (0.557 − 0.830i)7-s − 0.353·8-s + (0.543 − 0.452i)10-s + 1.08i·11-s + (−0.467 + 0.810i)13-s + (−0.311 − 0.634i)14-s + (−0.125 + 0.216i)16-s + (1.55 + 0.897i)17-s + (−0.953 + 0.550i)19-s + (−0.0853 − 0.492i)20-s + (0.666 + 0.384i)22-s + 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.966 + 0.255i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.624573133\)
\(L(\frac12)\) \(\approx\) \(2.624573133\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-2.09 - 0.770i)T \)
7 \( 1 + (-1.47 + 2.19i)T \)
good11 \( 1 - 3.61iT - 11T^{2} \)
13 \( 1 + (1.68 - 2.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.40 - 3.69i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.15 - 2.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.98T + 23T^{2} \)
29 \( 1 + (7.66 - 4.42i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.58 + 3.22i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.56 + 3.21i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.31 - 2.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.660 - 0.381i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.12 + 4.11i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.47 - 2.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.562 + 0.974i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.45 - 0.840i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.11 + 2.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.40iT - 71T^{2} \)
73 \( 1 + (2.78 - 4.81i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0862 - 0.149i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.5 + 6.65i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.08 + 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.18 + 7.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.645051135774914797892647575527, −8.503842283666255149224823534896, −7.46780463668151879646905913841, −6.82762432388839530082756260439, −5.87045293426815800772030756112, −4.98351783219694065750956984391, −4.26110109562285102889992707893, −3.25876745335394606498147752622, −2.02748139488138175864576955116, −1.38674768851392139614350542026, 0.955461048048482142889691793113, 2.52456406721506180964075413918, 3.21871611511843498506030669760, 4.78938444186511609045845072365, 5.28976581627276069363143730068, 5.87600563342635657536548537002, 6.70523113832559784963463254113, 7.81884638434090312230947390151, 8.368003880086082319179576495900, 9.181085832935970249792198012902

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