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Properties

Label 2-1900-95.64-c1-0-1
Degree $2$
Conductor $1900$
Sign $-0.786 - 0.617i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  + (0.315 − 0.182i)3-s + 0.635i·7-s + (−1.43 + 2.48i)9-s + 1.63·11-s + (−0.866 − 0.5i)13-s + (−5.71 + 3.29i)17-s + (−0.0466 − 4.35i)19-s + (0.115 + 0.200i)21-s + (0.750 + 0.433i)23-s + 2.13i·27-s + (−4.54 + 7.87i)29-s − 1.86·31-s + (0.516 − 0.298i)33-s + 0.635i·37-s − 0.364·39-s + ⋯
L(s)  = 1  + (0.182 − 0.105i)3-s + 0.240i·7-s + (−0.477 + 0.827i)9-s + 0.493·11-s + (−0.240 − 0.138i)13-s + (−1.38 + 0.799i)17-s + (−0.0107 − 0.999i)19-s + (0.0252 + 0.0437i)21-s + (0.156 + 0.0904i)23-s + 0.411i·27-s + (−0.844 + 1.46i)29-s − 0.335·31-s + (0.0898 − 0.0518i)33-s + 0.104i·37-s − 0.0583·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.786 - 0.617i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.786 - 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7107661664\)
\(L(\frac12)\) \(\approx\) \(0.7107661664\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + (0.0466 + 4.35i)T \)
good3 \( 1 + (-0.315 + 0.182i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.635iT - 7T^{2} \)
11 \( 1 - 1.63T + 11T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.71 - 3.29i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.750 - 0.433i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.54 - 7.87i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 - 0.635iT - 37T^{2} \)
41 \( 1 + (0.953 + 1.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.42 - 1.98i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.87 + 4.54i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.54 + 4.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.54 - 7.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.13 - 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.7 + 6.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.25 + 2.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.349 + 0.201i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.36 - 9.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 + (-0.271 + 0.469i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.38 - 3.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.236053896558225911859478610519, −8.821845920740890397886619855490, −8.059132497732350919029503445881, −7.11232331206850963310110008040, −6.47960107163018294369073267316, −5.40258925459166410457890409242, −4.74487747601036482277220688815, −3.64932513501897173128823069863, −2.58788960151925874124740143674, −1.68636311984860860137736944034, 0.23534179522943985335501346592, 1.81858888988176071684316409698, 2.97885442624987118996067369151, 3.92140537529493865189367152928, 4.65827555877246183919303718555, 5.85498894161292422849057811675, 6.48269714835033574374100510134, 7.31834737106669977806826827285, 8.186902052022556771656337034491, 9.061286331130605444256806815094

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