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Properties

Label 2-1890-63.47-c1-0-13
Degree $2$
Conductor $1890$
Sign $0.916 - 0.400i$
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 + (0.5 + 2.59i)7-s − 0.999i·8-s + (0.866 − 0.5i)10-s + 4.73i·11-s + (3 − 1.73i)13-s + (1.73 + 2i)14-s + (−0.5 − 0.866i)16-s + (−1.09 − 0.633i)19-s + (0.499 − 0.866i)20-s + (2.36 + 4.09i)22-s + 2.53i·23-s + 25-s + (1.73 − 3i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447·5-s + (0.188 + 0.981i)7-s − 0.353i·8-s + (0.273 − 0.158i)10-s + 1.42i·11-s + (0.832 − 0.480i)13-s + (0.462 + 0.534i)14-s + (−0.125 − 0.216i)16-s + (−0.251 − 0.145i)19-s + (0.111 − 0.193i)20-s + (0.504 + 0.873i)22-s + 0.528i·23-s + 0.200·25-s + (0.339 − 0.588i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.798863327\)
\(L(\frac12)\) \(\approx\) \(2.798863327\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good11 \( 1 - 4.73iT - 11T^{2} \)
13 \( 1 + (-3 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 + 0.633i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.53iT - 23T^{2} \)
29 \( 1 + (-5.59 - 3.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.09 + 4.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.09 - 5.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.59 - 2.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.29 + 3.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.09 - 1.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.1 + 6.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.73iT - 71T^{2} \)
73 \( 1 + (-6 + 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.29 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.59 - 9.69i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.19 + 3.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.39 - 4.26i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.420599090495072397572018922189, −8.597583155848577001665239834377, −7.67160705131600079575105688872, −6.67892349701237737293910871548, −5.94800620183781976543404625275, −5.17262104740355217558557403993, −4.46095499200997460443775368632, −3.29203906782188783976640306123, −2.33758094435102782176287282949, −1.47689164709364585491028535805, 0.885637081994300675766033364320, 2.29807041319235754367650654081, 3.60507474556560360480875699167, 4.04845525219767451301061746905, 5.26803686107743835592181688541, 5.92311555631396566986378582265, 6.73551558684861820641633432032, 7.38149183539610375713714053030, 8.583432563083270821902512209679, 8.744455895000956716354646035238

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