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Properties

Label 2-1900-19.7-c1-0-24
Degree $2$
Conductor $1900$
Sign $-0.0133 + 0.999i$
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  + (1.43 + 2.48i)3-s − 3.54·7-s + (−2.62 + 4.54i)9-s − 1.81·11-s + (1.60 − 2.78i)13-s + (−3.99 − 6.92i)17-s + (−0.863 − 4.27i)19-s + (−5.09 − 8.82i)21-s + (−4.21 + 7.30i)23-s − 6.46·27-s + (4.29 − 7.43i)29-s − 1.70·31-s + (−2.60 − 4.51i)33-s − 5.50·37-s + 9.23·39-s + ⋯
L(s)  = 1  + (0.829 + 1.43i)3-s − 1.34·7-s + (−0.875 + 1.51i)9-s − 0.547·11-s + (0.445 − 0.771i)13-s + (−0.969 − 1.67i)17-s + (−0.198 − 0.980i)19-s + (−1.11 − 1.92i)21-s + (−0.878 + 1.52i)23-s − 1.24·27-s + (0.796 − 1.38i)29-s − 0.306·31-s + (−0.453 − 0.786i)33-s − 0.905·37-s + 1.47·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0133 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0133 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.0133 + 0.999i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.0133 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3717494356\)
\(L(\frac12)\) \(\approx\) \(0.3717494356\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.863 + 4.27i)T \)
good3 \( 1 + (-1.43 - 2.48i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 + 1.81T + 11T^{2} \)
13 \( 1 + (-1.60 + 2.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.99 + 6.92i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.21 - 7.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.29 + 7.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + 5.50T + 37T^{2} \)
41 \( 1 + (-4.05 - 7.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.51 + 4.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.674 - 1.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.10 - 1.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.960 + 1.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.83 + 4.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.64 - 8.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.94 + 5.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.63 - 2.82i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.08 + 3.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.30T + 83T^{2} \)
89 \( 1 + (2.73 - 4.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.99 - 6.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.279489480809386431280127913519, −8.471974980118081478421881731796, −7.60578136918526779827729309045, −6.63930842089068886132530038936, −5.61695882964196553178670331373, −4.79569882909819742905471852405, −3.92235025837724925695358501584, −3.05909162519151272202847045645, −2.57396455731218973048031639074, −0.11158201883957409702420438940, 1.56771897758567467692143218410, 2.38057181550254575973775279441, 3.36724367374090822779954432075, 4.19150494320977574801906767216, 5.85111370720168866857992001279, 6.54555469135610787873556310653, 6.83442534366128022592779192241, 7.952633411666204636706273293306, 8.579339395352379937597656565894, 9.025684782693828517009198402114

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