Properties

Label 2-1900-19.18-c2-0-41
Degree $2$
Conductor $1900$
Sign $-0.568 + 0.822i$
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  − 3.42i·3-s − 9.18·7-s − 2.73·9-s + 11.4·11-s + 10.1i·13-s + 4.19·17-s + (10.8 − 15.6i)19-s + 31.4i·21-s + 0.952·23-s − 21.4i·27-s − 9.43i·29-s + 8.52i·31-s − 39.1i·33-s − 20.0i·37-s + 34.6·39-s + ⋯
L(s)  = 1  − 1.14i·3-s − 1.31·7-s − 0.303·9-s + 1.03·11-s + 0.778i·13-s + 0.246·17-s + (0.568 − 0.822i)19-s + 1.49i·21-s + 0.0414·23-s − 0.795i·27-s − 0.325i·29-s + 0.275i·31-s − 1.18i·33-s − 0.540i·37-s + 0.889·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.568 + 0.822i$
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: \((2,\ 1900,\ (\ :1),\ -0.568 + 0.822i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.595445573\)
\(L(\frac12)\) \(\approx\) \(1.595445573\)
\(L(2)\) 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 \( 1 + (-10.8 + 15.6i)T \)
good3 \( 1 + 3.42iT - 9T^{2} \)
7 \( 1 + 9.18T + 49T^{2} \)
11 \( 1 - 11.4T + 121T^{2} \)
13 \( 1 - 10.1iT - 169T^{2} \)
17 \( 1 - 4.19T + 289T^{2} \)
23 \( 1 - 0.952T + 529T^{2} \)
29 \( 1 + 9.43iT - 841T^{2} \)
31 \( 1 - 8.52iT - 961T^{2} \)
37 \( 1 + 20.0iT - 1.36e3T^{2} \)
41 \( 1 - 61.2iT - 1.68e3T^{2} \)
43 \( 1 - 57.9T + 1.84e3T^{2} \)
47 \( 1 - 35.3T + 2.20e3T^{2} \)
53 \( 1 + 57.6iT - 2.80e3T^{2} \)
59 \( 1 + 16.7iT - 3.48e3T^{2} \)
61 \( 1 + 32.2T + 3.72e3T^{2} \)
67 \( 1 - 6.07iT - 4.48e3T^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 - 27.9T + 5.32e3T^{2} \)
79 \( 1 + 65.7iT - 6.24e3T^{2} \)
83 \( 1 + 60.2T + 6.88e3T^{2} \)
89 \( 1 + 97.9iT - 7.92e3T^{2} \)
97 \( 1 + 92.0iT - 9.40e3T^{2} \)
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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

−8.900385575469164069204365865280, −7.73344419362170581960471655799, −7.06518646021704990991614682461, −6.49227796819244999925486508501, −5.96763022996207258976304472286, −4.59395154753339811820481003823, −3.61682645692307902391843578339, −2.64299050239022561270567925858, −1.52311077650644271270507602034, −0.48864343439614198060089530918, 1.06001296138995302287904806363, 2.77073338386557353831792377900, 3.66338403221633536945825237986, 4.08763732606172822171135323475, 5.32953477259042143547049723639, 5.97394830189443966651556211285, 6.88513806567456164047337131324, 7.71451331038889224812896909175, 8.928001048588117598795416473624, 9.324618691803068771495990275216

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