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Properties

Label 2-1900-19.11-c1-0-10
Degree $2$
Conductor $1900$
Sign $0.296 - 0.954i$
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  + (0.0695 − 0.120i)3-s + 3.40·7-s + (1.49 + 2.58i)9-s + 0.185·11-s + (1.17 + 2.03i)13-s + (−3.35 + 5.81i)17-s + (−2.14 + 3.79i)19-s + (0.236 − 0.410i)21-s + (0.463 + 0.803i)23-s + 0.831·27-s + (0.677 + 1.17i)29-s − 9.46·31-s + (0.0129 − 0.0223i)33-s + 1.62·37-s + 0.327·39-s + ⋯
L(s)  = 1  + (0.0401 − 0.0695i)3-s + 1.28·7-s + (0.496 + 0.860i)9-s + 0.0559·11-s + (0.326 + 0.565i)13-s + (−0.814 + 1.41i)17-s + (−0.491 + 0.870i)19-s + (0.0516 − 0.0895i)21-s + (0.0967 + 0.167i)23-s + 0.160·27-s + (0.125 + 0.217i)29-s − 1.69·31-s + (0.00224 − 0.00389i)33-s + 0.267·37-s + 0.0524·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.296 - 0.954i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.906447230\)
\(L(\frac12)\) \(\approx\) \(1.906447230\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 3.79i)T \)
good3 \( 1 + (-0.0695 + 0.120i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 3.40T + 7T^{2} \)
11 \( 1 - 0.185T + 11T^{2} \)
13 \( 1 + (-1.17 - 2.03i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.35 - 5.81i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.463 - 0.803i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.677 - 1.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.46T + 31T^{2} \)
37 \( 1 - 1.62T + 37T^{2} \)
41 \( 1 + (-4.29 + 7.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.80 - 6.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.09 + 7.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.70 - 11.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.02 + 3.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.31 + 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.62 - 2.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.59 + 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.45 - 4.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.98 - 5.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.30T + 83T^{2} \)
89 \( 1 + (-1.95 - 3.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.25 - 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.222175432987498036196851878991, −8.442364915278389109204593362730, −7.932586489623751316040602309780, −7.13230394639341889862914011965, −6.17021219872377247181618320299, −5.27188525965866125668935061932, −4.42538050137035918189562172293, −3.76307053678162564391521824054, −2.02820894293664271557121471385, −1.63828493339445831929368176866, 0.69892016095727213703172930656, 1.96322140952955346287136559460, 3.09717895236831438095068366106, 4.26758686299951196234534901315, 4.85353264541865397289078194145, 5.79006559840040136034364677119, 6.86255034753032210056351140740, 7.39135901791822760141065277068, 8.396743203666375363840256035734, 8.992916585634637309120859177107

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