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Properties

Label 2-1890-315.104-c1-0-5
Degree $2$
Conductor $1890$
Sign $0.119 - 0.992i$
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 + (−2.16 − 0.548i)5-s + (−1.29 + 2.30i)7-s + 0.999·8-s + (0.609 + 2.15i)10-s + (2.30 − 1.33i)11-s + (−1.15 + 2.00i)13-s + (2.64 − 0.0312i)14-s + (−0.5 − 0.866i)16-s − 7.23i·17-s − 5.58i·19-s + (1.55 − 1.60i)20-s + (−2.30 − 1.33i)22-s + (−0.354 + 0.613i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.969 − 0.245i)5-s + (−0.489 + 0.871i)7-s + 0.353·8-s + (0.192 + 0.680i)10-s + (0.695 − 0.401i)11-s + (−0.321 + 0.557i)13-s + (0.707 − 0.00834i)14-s + (−0.125 − 0.216i)16-s − 1.75i·17-s − 1.28i·19-s + (0.348 − 0.358i)20-s + (−0.491 − 0.283i)22-s + (−0.0738 + 0.127i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.119 - 0.992i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.119 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4611872126\)
\(L(\frac12)\) \(\approx\) \(0.4611872126\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 0.548i)T \)
7 \( 1 + (1.29 - 2.30i)T \)
good11 \( 1 + (-2.30 + 1.33i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.15 - 2.00i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.23iT - 17T^{2} \)
19 \( 1 + 5.58iT - 19T^{2} \)
23 \( 1 + (0.354 - 0.613i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.68 + 1.55i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.32 - 3.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.09iT - 37T^{2} \)
41 \( 1 + (3.67 - 6.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (11.3 - 6.54i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.91 - 2.83i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.40T + 53T^{2} \)
59 \( 1 + (1.98 - 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.66 - 5.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.96 - 2.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.75iT - 71T^{2} \)
73 \( 1 - 4.33T + 73T^{2} \)
79 \( 1 + (-7.29 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.25 - 1.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.66T + 89T^{2} \)
97 \( 1 + (6.43 + 11.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.451953824329050647006093765405, −8.674184804283797927979499002756, −8.146384311940802760830295626336, −6.98829871969149965941750416240, −6.51627315401767918487909796077, −4.98772629429259283415775357085, −4.55424359304405004205655193138, −3.19157256819937645588323324566, −2.75959276183956658109713407854, −1.12489839916912793494704686522, 0.22317522399295051159447396579, 1.68877008597358718642750596666, 3.50462720521435352459931061870, 3.92629460563662064043162509956, 4.94619006770433728753396081216, 6.21437806714626835765295972252, 6.65809020084209463292738704030, 7.61383515279077747388701271240, 8.052120128157482490626056330024, 8.831620519376754271355481337905

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