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Properties

Label 2-1890-315.209-c1-0-20
Degree $2$
Conductor $1890$
Sign $0.0230 - 0.999i$
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 + (2.18 + 0.462i)5-s + (2.50 + 0.864i)7-s + 0.999·8-s + (−1.49 + 1.66i)10-s + (1.78 + 1.03i)11-s + (−1.79 − 3.11i)13-s + (−1.99 + 1.73i)14-s + (−0.5 + 0.866i)16-s + 6.40i·17-s + 3.01i·19-s + (−0.693 − 2.12i)20-s + (−1.78 + 1.03i)22-s + (2.09 + 3.62i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.978 + 0.206i)5-s + (0.945 + 0.326i)7-s + 0.353·8-s + (−0.472 + 0.526i)10-s + (0.537 + 0.310i)11-s + (−0.498 − 0.864i)13-s + (−0.534 + 0.463i)14-s + (−0.125 + 0.216i)16-s + 1.55i·17-s + 0.691i·19-s + (−0.155 − 0.475i)20-s + (−0.380 + 0.219i)22-s + (0.436 + 0.755i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0230 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.0230 - 0.999i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.0230 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.882018754\)
\(L(\frac12)\) \(\approx\) \(1.882018754\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 0.462i)T \)
7 \( 1 + (-2.50 - 0.864i)T \)
good11 \( 1 + (-1.78 - 1.03i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.79 + 3.11i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.40iT - 17T^{2} \)
19 \( 1 - 3.01iT - 19T^{2} \)
23 \( 1 + (-2.09 - 3.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.33 + 4.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.25 - 1.30i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.05iT - 37T^{2} \)
41 \( 1 + (-0.0615 - 0.106i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.10 - 2.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.79 - 2.76i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.92T + 53T^{2} \)
59 \( 1 + (6.54 + 11.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.524 + 0.302i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.0 + 6.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.91iT - 71T^{2} \)
73 \( 1 - 6.80T + 73T^{2} \)
79 \( 1 + (2.84 - 4.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.8 + 8.59i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.98T + 89T^{2} \)
97 \( 1 + (-0.320 + 0.555i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.410359458090672789242234553504, −8.567679424904939516282174590638, −7.85132204975655904409146546891, −7.16437859549702086893562635895, −6.00800189497599969565367412578, −5.70136386642392412325978932915, −4.78026962336965008074165006204, −3.64884884258939638366255740931, −2.16940257819400234108615847678, −1.40081886014490923071036213287, 0.839046612657666472135100804311, 1.92832782203040543071312620135, 2.72978280957759387613473209839, 4.10713249103638053212459143244, 4.88297038266634732674573981411, 5.62309349145958252103256127324, 6.96522687049804118051371320388, 7.31559452969353928898460201396, 8.635110978378146763516015907341, 9.112886715198358681480843129100

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