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Properties

Label 2-1890-63.59-c1-0-5
Degree $2$
Conductor $1890$
Sign $-0.599 - 0.800i$
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 + (2.20 − 1.46i)7-s + 0.999i·8-s + (−0.866 − 0.5i)10-s + 4.43i·11-s + (−4.51 − 2.60i)13-s + (2.64 − 0.172i)14-s + (−0.5 + 0.866i)16-s + (−3.46 + 6.00i)17-s + (−2.43 + 1.40i)19-s + (−0.499 − 0.866i)20-s + (−2.21 + 3.84i)22-s + 3.48i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.831 − 0.555i)7-s + 0.353i·8-s + (−0.273 − 0.158i)10-s + 1.33i·11-s + (−1.25 − 0.723i)13-s + (0.705 − 0.0460i)14-s + (−0.125 + 0.216i)16-s + (−0.841 + 1.45i)17-s + (−0.557 + 0.321i)19-s + (−0.111 − 0.193i)20-s + (−0.473 + 0.819i)22-s + 0.726i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.599 - 0.800i$
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: \((2,\ 1890,\ (\ :1/2),\ -0.599 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.710125236\)
\(L(\frac12)\) \(\approx\) \(1.710125236\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 1.46i)T \)
good11 \( 1 - 4.43iT - 11T^{2} \)
13 \( 1 + (4.51 + 2.60i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.46 - 6.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.43 - 1.40i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.48iT - 23T^{2} \)
29 \( 1 + (-6.40 + 3.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.82 - 1.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.36 - 7.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.90 - 5.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.02 + 1.77i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.16 - 8.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.60 - 2.65i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.534 - 0.926i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.23 - 3.02i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.46 + 4.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 + (-13.1 - 7.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.88 - 6.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.84 - 4.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.16 + 7.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.4 - 6.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.608822199218139727003478428596, −8.160331755264361403436787438971, −8.046788345519887396885489876327, −7.09488945149232930748406785161, −6.44763550999012100116816660363, −5.25198310156635986105995866136, −4.52711312749187124154988125003, −4.06178591355026411011600026010, −2.70763314215815185389260270936, −1.63434542251137836136112571185, 0.48221508608464041761149572489, 2.17805492886181089697482226963, 2.82437611670937606162167685558, 4.07621742479441603486421276162, 4.85798880516213837862118528879, 5.41606360843120802953214056675, 6.60838200621224516324450025896, 7.18572662069576028142866734067, 8.334129262829280477897204395680, 8.854366352946343022063670125405

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