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Properties

Label 2-1890-9.7-c1-0-0
Degree $2$
Conductor $1890$
Sign $-0.454 - 0.890i$
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 + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s − 0.999·8-s − 0.999·10-s + (0.367 − 0.635i)11-s + (0.867 + 1.50i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 2.52·17-s − 6.52·19-s + (−0.499 + 0.866i)20-s + (−0.367 − 0.635i)22-s + (−1.86 − 3.23i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.188 + 0.327i)7-s − 0.353·8-s − 0.316·10-s + (0.110 − 0.191i)11-s + (0.240 + 0.416i)13-s + (0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s − 0.612·17-s − 1.49·19-s + (−0.111 + 0.193i)20-s + (−0.0782 − 0.135i)22-s + (−0.389 − 0.674i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.454 - 0.890i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06792999668\)
\(L(\frac12)\) \(\approx\) \(0.06792999668\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (-0.367 + 0.635i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.867 - 1.50i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.52T + 17T^{2} \)
19 \( 1 + 6.52T + 19T^{2} \)
23 \( 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.13 - 1.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.25T + 37T^{2} \)
41 \( 1 + (3.36 + 5.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.79 - 8.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.25T + 53T^{2} \)
59 \( 1 + (-0.527 - 0.914i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.89 - 3.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.99 + 5.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.25T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + (6.52 - 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.79 - 13.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.04T + 89T^{2} \)
97 \( 1 + (0.234 - 0.405i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.425634928790352770985545813192, −8.724563117272846752971873094477, −8.247059807079917914744616501518, −6.87364439595419629285852367481, −6.29140769681051273818046664785, −5.28045178221490894886311019200, −4.43635023753078795128649979195, −3.72385784632142449826903249340, −2.57742882581569824596665357230, −1.59653001791727190839445598568, 0.02069648646090077801819323904, 1.98739834285154253024674566291, 3.24315116556868589948261726344, 4.05061838846299623753779630566, 4.84173638326109957623870808935, 5.95218472447147216285183940941, 6.55857318165886860148441834856, 7.27821470643721026897020235290, 8.124897946909885246974302734965, 8.720350453491510969268537194683

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