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Properties

Label 2-1890-63.59-c1-0-30
Degree $2$
Conductor $1890$
Sign $-0.948 - 0.318i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + 5-s + (−0.555 − 2.58i)7-s − 0.999i·8-s + (−0.866 − 0.5i)10-s + 4.07i·11-s + (−5.20 − 3.00i)13-s + (−0.812 + 2.51i)14-s + (−0.5 + 0.866i)16-s + (0.641 − 1.11i)17-s + (2.90 − 1.67i)19-s + (0.499 + 0.866i)20-s + (2.03 − 3.53i)22-s − 2.18i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.209 − 0.977i)7-s − 0.353i·8-s + (−0.273 − 0.158i)10-s + 1.22i·11-s + (−1.44 − 0.832i)13-s + (−0.217 + 0.672i)14-s + (−0.125 + 0.216i)16-s + (0.155 − 0.269i)17-s + (0.666 − 0.385i)19-s + (0.111 + 0.193i)20-s + (0.434 − 0.752i)22-s − 0.456i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.948 - 0.318i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.948 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1899411920\)
\(L(\frac12)\) \(\approx\) \(0.1899411920\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + (0.555 + 2.58i)T \)
good11 \( 1 - 4.07iT - 11T^{2} \)
13 \( 1 + (5.20 + 3.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.641 + 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.90 + 1.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.18iT - 23T^{2} \)
29 \( 1 + (6.21 - 3.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.79 - 3.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.90 + 5.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.36 - 2.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.43 + 4.22i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.13 - 7.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.48 - 4.32i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.630 - 1.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.56 + 1.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.31 - 7.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (6.02 + 3.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.09 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.87 + 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.29 - 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.8 - 9.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.180384317936733111318463169751, −7.78666878781023421097961305373, −7.32876594928518306119455764439, −6.81324956665194515023753311196, −5.41475218504796961945944371445, −4.72324498441545999804349220180, −3.57349924391848451945754738717, −2.58890792468523633930979225740, −1.53539784825929101877439775414, −0.080309760732541577824539539953, 1.69977973929405771580457036990, 2.61264324534951585232531547574, 3.74894490281287869746755713295, 5.28143696786866422602163078414, 5.59183695863479723338584256700, 6.52912302254395043336180111416, 7.32094669921853092041188555212, 8.213290874708642635196037554388, 8.893591335063939310727073875155, 9.665167607551694044290849348311

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