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Properties

Label 2-1890-315.104-c1-0-39
Degree $2$
Conductor $1890$
Sign $-0.878 + 0.477i$
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 + (1.00 − 1.99i)5-s + (−2.07 + 1.64i)7-s + 0.999·8-s + (−2.23 + 0.124i)10-s + (1.44 − 0.832i)11-s + (1.04 − 1.81i)13-s + (2.45 + 0.977i)14-s + (−0.5 − 0.866i)16-s − 7.45i·17-s + 4.66i·19-s + (1.22 + 1.87i)20-s + (−1.44 − 0.832i)22-s + (−2.82 + 4.89i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.450 − 0.892i)5-s + (−0.784 + 0.619i)7-s + 0.353·8-s + (−0.706 + 0.0394i)10-s + (0.434 − 0.250i)11-s + (0.290 − 0.503i)13-s + (0.657 + 0.261i)14-s + (−0.125 − 0.216i)16-s − 1.80i·17-s + 1.06i·19-s + (0.273 + 0.418i)20-s + (−0.307 − 0.177i)22-s + (−0.589 + 1.02i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.878 + 0.477i$
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: \((2,\ 1890,\ (\ :1/2),\ -0.878 + 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007911014\)
\(L(\frac12)\) \(\approx\) \(1.007911014\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.00 + 1.99i)T \)
7 \( 1 + (2.07 - 1.64i)T \)
good11 \( 1 + (-1.44 + 0.832i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.04 + 1.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.45iT - 17T^{2} \)
19 \( 1 - 4.66iT - 19T^{2} \)
23 \( 1 + (2.82 - 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.10 + 2.37i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.73 - 1.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.86iT - 37T^{2} \)
41 \( 1 + (-4.07 + 7.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.25 + 1.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.90 - 5.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 + (-0.0862 + 0.149i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.17 + 2.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.70 + 2.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 - 4.79T + 73T^{2} \)
79 \( 1 + (6.30 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.24 + 1.29i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-4.73 - 8.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.215262880753903733879435472606, −8.293844926720708929815789982170, −7.57211042836310819632478522735, −6.34399673186259581767800590447, −5.66686522352312443672343123261, −4.80484455978086626260072546986, −3.67517700973229900973322109158, −2.80994483177969262280932594915, −1.68076904927100001056859765163, −0.43043126269365501877206842611, 1.39572853489826057234454051162, 2.70772999736752368421163670741, 3.81154648739988189936452403357, 4.62791337747764427681973638907, 6.02224798954763949988594246729, 6.52656362854287787588016681154, 6.88374640764767973343583873522, 7.981675276633850064770721566169, 8.703532153136559755185253229684, 9.679430299567629682288168977167

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