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Properties

Label 2-1890-315.104-c1-0-36
Degree $2$
Conductor $1890$
Sign $0.999 - 0.0135i$
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 + (0.150 + 2.23i)5-s + (−2.28 − 1.32i)7-s − 0.999·8-s + (−1.85 + 1.24i)10-s + (2.86 − 1.65i)11-s + (2.91 − 5.05i)13-s + (0.00309 − 2.64i)14-s + (−0.5 − 0.866i)16-s − 2.78i·17-s + 0.0210i·19-s + (−2.00 − 0.984i)20-s + (2.86 + 1.65i)22-s + (2.80 − 4.85i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0674 + 0.997i)5-s + (−0.865 − 0.501i)7-s − 0.353·8-s + (−0.587 + 0.394i)10-s + (0.865 − 0.499i)11-s + (0.809 − 1.40i)13-s + (0.000827 − 0.707i)14-s + (−0.125 − 0.216i)16-s − 0.674i·17-s + 0.00483i·19-s + (−0.448 − 0.220i)20-s + (0.611 + 0.353i)22-s + (0.583 − 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0135i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.999 - 0.0135i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.775338592\)
\(L(\frac12)\) \(\approx\) \(1.775338592\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.150 - 2.23i)T \)
7 \( 1 + (2.28 + 1.32i)T \)
good11 \( 1 + (-2.86 + 1.65i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.91 + 5.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.78iT - 17T^{2} \)
19 \( 1 - 0.0210iT - 19T^{2} \)
23 \( 1 + (-2.80 + 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.43 - 0.828i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.24 + 3.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.54iT - 37T^{2} \)
41 \( 1 + (-5.58 + 9.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.13 + 1.23i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.89 + 2.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.154T + 53T^{2} \)
59 \( 1 + (-3.32 + 5.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.58 - 4.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.9 - 6.89i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + 0.567T + 73T^{2} \)
79 \( 1 + (-2.02 - 3.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.5 + 6.66i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.33T + 89T^{2} \)
97 \( 1 + (4.93 + 8.54i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.148822277233991292358345722467, −8.351071061088146651696900725202, −7.34309026324695458808547575435, −6.87662053443733573260929496738, −6.07031927370294931612387067538, −5.50925325211850187470477878800, −4.05950893138784620222284777421, −3.45313156785415977680135655197, −2.66607633168060673113692153531, −0.64227433289911072051500375950, 1.25382922013063988574152667673, 2.07568894121087231111937122758, 3.54535784622866820297161810329, 4.09888413525028463660529207659, 5.04499038067542626507974179787, 6.01502323927314333308882039324, 6.55041181796231786598139110190, 7.70843980456914660838455553007, 9.027309630336741359730061747691, 9.098416248844371778587495144337

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