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Properties

Label 2-1890-315.59-c1-0-23
Degree $2$
Conductor $1890$
Sign $0.543 + 0.839i$
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 + (−2.07 − 0.824i)5-s + (0.837 + 2.50i)7-s − 0.999·8-s + (−1.75 + 1.38i)10-s + 0.811i·11-s + (−0.0126 + 0.0219i)13-s + (2.59 + 0.529i)14-s + (−0.5 + 0.866i)16-s + (−2.78 − 1.60i)17-s + (1.22 − 0.705i)19-s + (0.325 + 2.21i)20-s + (0.702 + 0.405i)22-s + 3.53·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.929 − 0.368i)5-s + (0.316 + 0.948i)7-s − 0.353·8-s + (−0.554 + 0.438i)10-s + 0.244i·11-s + (−0.00351 + 0.00609i)13-s + (0.692 + 0.141i)14-s + (−0.125 + 0.216i)16-s + (−0.675 − 0.390i)17-s + (0.280 − 0.161i)19-s + (0.0728 + 0.494i)20-s + (0.149 + 0.0864i)22-s + 0.738·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.543 + 0.839i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.682708175\)
\(L(\frac12)\) \(\approx\) \(1.682708175\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 0.824i)T \)
7 \( 1 + (-0.837 - 2.50i)T \)
good11 \( 1 - 0.811iT - 11T^{2} \)
13 \( 1 + (0.0126 - 0.0219i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.78 + 1.60i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.22 + 0.705i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 + (-4.38 + 2.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.89 + 1.67i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.76 + 5.63i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.973 - 1.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.76 - 1.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.50 - 3.75i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.33 + 9.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.80 - 3.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.527 - 0.304i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.879 + 0.507i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (-6.25 + 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.845 - 1.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-14.5 + 8.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.31 + 4.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.19 + 5.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.097684062123442221823008743292, −8.450531012421419275309673630194, −7.65549683885824867318914896626, −6.67735316367538619030178352065, −5.66548873309372926226325276898, −4.80140616608381544661413372895, −4.25190367991087926132819654756, −3.07217505949124219894710640057, −2.24740429886541718331502467517, −0.789199014015520244690719397833, 0.897166802191563192832336880967, 2.75166336921238299399195505540, 3.72865397756122005154523610433, 4.39399460547639949152916324132, 5.17526517335176905136251201667, 6.45383794648058849715966910256, 6.90768903041808041634694367846, 7.76192440370881229274153248023, 8.265466878652907883816445647034, 9.111138655561655096144274652216

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