Properties

Label 2-1900-19.18-c2-0-59
Degree $2$
Conductor $1900$
Sign $-0.613 + 0.790i$
Analytic cond. $51.7712$
Root an. cond. $7.19522$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.630i·3-s + 3.98·7-s + 8.60·9-s − 5.54·11-s − 23.1i·13-s − 16.5·17-s + (11.6 − 15.0i)19-s − 2.51i·21-s − 6.86·23-s − 11.0i·27-s + 37.0i·29-s + 11.1i·31-s + 3.49i·33-s − 66.6i·37-s − 14.5·39-s + ⋯
L(s)  = 1  − 0.210i·3-s + 0.569·7-s + 0.955·9-s − 0.504·11-s − 1.77i·13-s − 0.970·17-s + (0.613 − 0.790i)19-s − 0.119i·21-s − 0.298·23-s − 0.410i·27-s + 1.27i·29-s + 0.358i·31-s + 0.105i·33-s − 1.80i·37-s − 0.373·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.613 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.613 + 0.790i$
Analytic conductor: \(51.7712\)
Root analytic conductor: \(7.19522\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1),\ -0.613 + 0.790i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.490629480\)
\(L(\frac12)\) \(\approx\) \(1.490629480\)
\(L(2)\) 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 \( 1 + (-11.6 + 15.0i)T \)
good3 \( 1 + 0.630iT - 9T^{2} \)
7 \( 1 - 3.98T + 49T^{2} \)
11 \( 1 + 5.54T + 121T^{2} \)
13 \( 1 + 23.1iT - 169T^{2} \)
17 \( 1 + 16.5T + 289T^{2} \)
23 \( 1 + 6.86T + 529T^{2} \)
29 \( 1 - 37.0iT - 841T^{2} \)
31 \( 1 - 11.1iT - 961T^{2} \)
37 \( 1 + 66.6iT - 1.36e3T^{2} \)
41 \( 1 + 14.8iT - 1.68e3T^{2} \)
43 \( 1 + 34.7T + 1.84e3T^{2} \)
47 \( 1 + 63.0T + 2.20e3T^{2} \)
53 \( 1 - 36.8iT - 2.80e3T^{2} \)
59 \( 1 + 19.0iT - 3.48e3T^{2} \)
61 \( 1 - 34.3T + 3.72e3T^{2} \)
67 \( 1 + 27.2iT - 4.48e3T^{2} \)
71 \( 1 - 90.8iT - 5.04e3T^{2} \)
73 \( 1 - 19.7T + 5.32e3T^{2} \)
79 \( 1 + 111. iT - 6.24e3T^{2} \)
83 \( 1 - 129.T + 6.88e3T^{2} \)
89 \( 1 + 66.5iT - 7.92e3T^{2} \)
97 \( 1 + 19.0iT - 9.40e3T^{2} \)
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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

−8.665568003676176775896412079326, −7.86292089450175076008318374224, −7.32024007704375864404892739223, −6.47524955430377874599744901281, −5.31052405959850723045603817762, −4.88067462587708987101144603904, −3.69490740195179558207791593746, −2.69021600969195035803814488642, −1.57404680223738504055777917030, −0.37398247647206728823353445724, 1.44314171200655227270169908555, 2.20912981020746269994436537356, 3.65464589476509515754156414423, 4.48460723350183078772458360552, 5.00039761705662766062588601306, 6.32762984703669409134274192290, 6.84342969083432647502251557107, 7.86165333023924527360047836861, 8.410508139007485606924758731540, 9.592765877120268854154583419741

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