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Properties

Label 2-1900-19.11-c1-0-25
Degree $2$
Conductor $1900$
Sign $-0.466 + 0.884i$
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  + (0.626 − 1.08i)3-s − 2.18·7-s + (0.714 + 1.23i)9-s + 2.12·11-s + (−2.58 − 4.46i)13-s + (−1.31 + 2.27i)17-s + (2.80 − 3.33i)19-s + (−1.36 + 2.37i)21-s + (−1.27 − 2.20i)23-s + 5.55·27-s + (−3.08 − 5.33i)29-s − 1.05·31-s + (1.33 − 2.31i)33-s − 2.25·37-s − 6.46·39-s + ⋯
L(s)  = 1  + (0.361 − 0.626i)3-s − 0.825·7-s + (0.238 + 0.412i)9-s + 0.642·11-s + (−0.715 − 1.23i)13-s + (−0.318 + 0.551i)17-s + (0.643 − 0.765i)19-s + (−0.298 + 0.517i)21-s + (−0.265 − 0.459i)23-s + 1.06·27-s + (−0.571 − 0.990i)29-s − 0.188·31-s + (0.232 − 0.402i)33-s − 0.371·37-s − 1.03·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.466 + 0.884i$
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: \((2,\ 1900,\ (\ :1/2),\ -0.466 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.379166884\)
\(L(\frac12)\) \(\approx\) \(1.379166884\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.80 + 3.33i)T \)
good3 \( 1 + (-0.626 + 1.08i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 2.18T + 7T^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 + (2.58 + 4.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.31 - 2.27i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.27 + 2.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.08 + 5.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.05T + 31T^{2} \)
37 \( 1 + 2.25T + 37T^{2} \)
41 \( 1 + (-5.02 + 8.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.840 - 1.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.24 + 5.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.63 + 6.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.53 + 6.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.41 - 9.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.41 - 2.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.11 - 5.39i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.78 + 13.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.80 + 4.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.55T + 83T^{2} \)
89 \( 1 + (7.73 + 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.63 + 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.881620929690291529188268695563, −8.104933926046520258643146719693, −7.35551498044227127695016369056, −6.75769403493184340586238454402, −5.85107401903375057167336609154, −4.93428210678080798939029162668, −3.82139077918655406399215018804, −2.86115615309815611709702363528, −1.96896283497060979164886587653, −0.48062106478968882160833833337, 1.44784350676261394214900025568, 2.83930576236876655058685360687, 3.69820345416446692840200418889, 4.35819648459751630345108049785, 5.36577659485015959752244013696, 6.52760687818189000409977358059, 6.91757306160525863452050286663, 7.939539498151482376812999534109, 9.072744624594861292552609092160, 9.532118580491917732958283042431

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