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Properties

Label 2-1890-315.209-c1-0-26
Degree $2$
Conductor $1890$
Sign $0.984 + 0.173i$
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 + (−1.25 + 1.84i)5-s + (1.02 + 2.43i)7-s + 0.999·8-s + (−0.973 − 2.01i)10-s + (2.49 + 1.44i)11-s + (−3.34 − 5.79i)13-s + (−2.62 − 0.331i)14-s + (−0.5 + 0.866i)16-s − 2.78i·17-s − 6.90i·19-s + (2.23 + 0.163i)20-s + (−2.49 + 1.44i)22-s + (−2.70 − 4.67i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.561 + 0.827i)5-s + (0.387 + 0.921i)7-s + 0.353·8-s + (−0.307 − 0.636i)10-s + (0.753 + 0.434i)11-s + (−0.928 − 1.60i)13-s + (−0.701 − 0.0884i)14-s + (−0.125 + 0.216i)16-s − 0.675i·17-s − 1.58i·19-s + (0.498 + 0.0364i)20-s + (−0.532 + 0.307i)22-s + (−0.563 − 0.975i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.984 + 0.173i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.012276011\)
\(L(\frac12)\) \(\approx\) \(1.012276011\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 - 1.84i)T \)
7 \( 1 + (-1.02 - 2.43i)T \)
good11 \( 1 + (-2.49 - 1.44i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.34 + 5.79i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.78iT - 17T^{2} \)
19 \( 1 + 6.90iT - 19T^{2} \)
23 \( 1 + (2.70 + 4.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.21 - 1.85i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.83 + 3.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.07iT - 37T^{2} \)
41 \( 1 + (0.0641 + 0.111i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.156 + 0.0905i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.40 - 2.54i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (5.33 + 9.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.13 + 3.54i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.2 + 7.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.02iT - 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + (-5.64 + 9.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.452 - 0.260i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.69T + 89T^{2} \)
97 \( 1 + (2.06 - 3.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.189360799789647854408846943177, −8.124500239299948365289784801762, −7.81732460740679008417711339754, −6.76713650880170574059299061122, −6.31502597257141249211482241501, −5.08278390275811198921125773333, −4.59486052687303406340869376873, −3.07297162579761062692490930880, −2.38416280028264534148934133325, −0.48426188057305154093045347966, 1.13087111634668105156222866290, 1.90297776950006637418684103649, 3.63992193398812259048888216478, 4.10163371925050656340025406920, 4.83723893730874810317276804142, 6.10003937701597226453029478548, 7.11042315845729585708915694917, 7.88511619875066717737575658643, 8.452755862058275770886325848886, 9.348539878613734291068894235619

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