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Properties

Label 2-1900-95.49-c1-0-13
Degree $2$
Conductor $1900$
Sign $0.966 + 0.258i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.966 + 0.258i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.966 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.640806286\)
\(L(\frac12)\) \(\approx\) \(1.640806286\)
\(L(\frac{3}{2})\) 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

−9.245219942087989179700404715810, −8.330425384976651021454557495004, −7.74646080933123079700935848052, −6.67935059070571904508650232207, −5.87720974537912356501770177941, −5.48348434698063752416962946534, −3.93231355760966792317364551341, −3.52075346631649825965628956060, −2.11719088183230713481288392009, −0.855131959933540068718877910492, 0.935637067458283935612006885027, 2.33696404648450825585670993453, 3.35635938730843539668441371038, 4.35933786376108017604305874002, 5.32280049244943772372465137051, 5.83956075307075801477126963358, 7.17184965137611838665981891367, 7.42268162730340513942741013913, 8.632371440206708136137935673606, 9.121344301284111414309166098662

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