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Properties

Label 2-1890-315.59-c1-0-33
Degree $2$
Conductor $1890$
Sign $0.268 + 0.963i$
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 + (−0.782 + 2.09i)5-s + (2.24 − 1.40i)7-s − 0.999·8-s + (1.42 + 1.72i)10-s + 0.274i·11-s + (1.02 − 1.77i)13-s + (−0.0934 − 2.64i)14-s + (−0.5 + 0.866i)16-s + (−3.20 − 1.84i)17-s + (−1.01 + 0.586i)19-s + (2.20 − 0.369i)20-s + (0.237 + 0.137i)22-s + 1.10·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.349 + 0.936i)5-s + (0.847 − 0.530i)7-s − 0.353·8-s + (0.449 + 0.545i)10-s + 0.0826i·11-s + (0.284 − 0.492i)13-s + (−0.0249 − 0.706i)14-s + (−0.125 + 0.216i)16-s + (−0.776 − 0.448i)17-s + (−0.232 + 0.134i)19-s + (0.493 − 0.0826i)20-s + (0.0506 + 0.0292i)22-s + 0.230·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.268 + 0.963i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.010438326\)
\(L(\frac12)\) \(\approx\) \(2.010438326\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.782 - 2.09i)T \)
7 \( 1 + (-2.24 + 1.40i)T \)
good11 \( 1 - 0.274iT - 11T^{2} \)
13 \( 1 + (-1.02 + 1.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.20 + 1.84i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.01 - 0.586i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.10T + 23T^{2} \)
29 \( 1 + (-8.83 + 5.10i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.50 + 2.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.32 + 2.49i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.504 + 0.873i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.578 + 0.334i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.16 - 5.29i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.31 - 7.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.49 + 4.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.78 - 4.49i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.227 + 0.131i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 + (0.316 - 0.548i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.59 + 11.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.58 - 3.22i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.16 + 12.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.30 - 9.18i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.149213162699131361612094202198, −8.165601611160570878377983318280, −7.56986326726595878123216267953, −6.61723358904802959560413453442, −5.87135517591589320103560060486, −4.61756192864254480961505612851, −4.17512598016905084017221896387, −3.01488028546641830811734961006, −2.24140048735349619149069211917, −0.76905585282502082824668135086, 1.18296338693865018776373279497, 2.50227375870166904994520028648, 3.88670998091974890097822863017, 4.66588387856320473606173311874, 5.17534981326803447885023005646, 6.18219656137008307516448892348, 6.94738566835897812718052934054, 7.987527983264203141014802785361, 8.618314653733349665781303467840, 8.873067943433801998566103661017

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