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Properties

Label 2-1900-95.83-c0-0-0
Degree $2$
Conductor $1900$
Sign $0.181 - 0.983i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.36i)3-s + (−0.866 − 0.5i)9-s + 11-s + (1.36 − 0.366i)13-s + (1.36 + 0.366i)17-s + (−0.866 − 0.5i)19-s + (0.866 + 0.5i)29-s − 31-s + (−0.366 + 1.36i)33-s + 2i·39-s + (−0.366 − 1.36i)47-s + i·49-s + (−1 + 1.73i)51-s + (−1.36 + 0.366i)53-s + (1 − 0.999i)57-s + ⋯
L(s)  = 1  + (−0.366 + 1.36i)3-s + (−0.866 − 0.5i)9-s + 11-s + (1.36 − 0.366i)13-s + (1.36 + 0.366i)17-s + (−0.866 − 0.5i)19-s + (0.866 + 0.5i)29-s − 31-s + (−0.366 + 1.36i)33-s + 2i·39-s + (−0.366 − 1.36i)47-s + i·49-s + (−1 + 1.73i)51-s + (−1.36 + 0.366i)53-s + (1 − 0.999i)57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.181 - 0.983i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 0.181 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.147690883\)
\(L(\frac12)\) \(\approx\) \(1.147690883\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
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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

−9.672336898887715887034447195152, −8.860237776783265919124476869259, −8.345201078015279357477258761190, −7.12920182589225668592762831817, −6.14369507817524753321458064786, −5.56752168242891846340404219303, −4.57207734016125532187622458314, −3.83237478187250010527826870355, −3.19533047471303788907302641488, −1.40333137115358951661151103741, 1.11136037456788912771054767476, 1.85360296171627117780084992564, 3.29790535915197883030244404753, 4.20755450339506852813172519535, 5.52724340308030184328501836729, 6.30958731717651397214464197458, 6.66012725687924664387944641000, 7.67169733773729531556472998695, 8.249844400221685003455104153863, 9.095340354317122819879183831186

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