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Properties

Label 2-1890-63.4-c1-0-4
Degree $2$
Conductor $1890$
Sign $-0.979 - 0.202i$
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 + 5-s + (2.63 − 0.236i)7-s + 0.999·8-s + (−0.5 + 0.866i)10-s − 1.64·11-s + (−2.49 + 4.31i)13-s + (−1.11 + 2.40i)14-s + (−0.5 + 0.866i)16-s + (−3.52 + 6.10i)17-s + (−3.51 − 6.08i)19-s + (−0.499 − 0.866i)20-s + (0.821 − 1.42i)22-s − 7.45·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.995 − 0.0895i)7-s + 0.353·8-s + (−0.158 + 0.273i)10-s − 0.495·11-s + (−0.691 + 1.19i)13-s + (−0.297 + 0.641i)14-s + (−0.125 + 0.216i)16-s + (−0.855 + 1.48i)17-s + (−0.805 − 1.39i)19-s + (−0.111 − 0.193i)20-s + (0.175 − 0.303i)22-s − 1.55·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.979 - 0.202i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.979 - 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7291074599\)
\(L(\frac12)\) \(\approx\) \(0.7291074599\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (-2.63 + 0.236i)T \)
good11 \( 1 + 1.64T + 11T^{2} \)
13 \( 1 + (2.49 - 4.31i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.52 - 6.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.51 + 6.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.45T + 23T^{2} \)
29 \( 1 + (-2.86 - 4.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.82 + 3.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.87 + 4.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.30 - 7.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.96 - 5.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.820 + 1.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.46 - 11.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.391 - 0.677i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.38 - 4.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.97 - 8.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.26T + 71T^{2} \)
73 \( 1 + (0.392 - 0.680i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.93 - 5.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.28 + 5.69i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.63 + 8.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.99 - 5.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.385430831995199012094853763365, −8.705030323071220454667433194265, −8.115542897742946958997703483844, −7.20695893544122519421931503349, −6.49946737126071153571570399144, −5.71598043446405596922170145662, −4.64637207280696834138494185571, −4.24810170713254275585567728810, −2.38259880185996743997306696930, −1.64993828540503621049367821456, 0.27632315600598750635430723016, 1.88496885960269559923359497012, 2.49828729543172165085646707798, 3.72824143915486076541191220406, 4.84455489376205504907321019989, 5.38645327557498624336802453952, 6.49123342056558144251016747042, 7.64848871842360720871676153361, 8.092171882650582220913812803056, 8.831280402258797989216807927309

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