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Properties

Label 2-1900-19.11-c1-0-15
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

\(L(1)\) \(\approx\) \(1.936513212\)
\(L(\frac12)\) \(\approx\) \(1.936513212\)
\(L(\frac{3}{2})\) 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

−9.300350181160053773475244145627, −8.809381395252047758787707316159, −7.64452708513286717286092194713, −7.13106517225101508003655612635, −5.99976735670484852413346505785, −5.23570523113489236106271146998, −4.39566457346083025121166255300, −3.83406306697491826668288548122, −2.33770681158590106169457694659, −1.21805961167911897570330692088, 0.934536528265668248359509595824, 1.67681808928030511402484139331, 3.23121919963739585763413847840, 4.03535816237555580240449129229, 5.23213229477094619176342979192, 5.91528367530633856395877626239, 6.65822384040333022601354155117, 7.60046006853780249421093223379, 8.128158095092513053584503114681, 8.982479223232768814827781582940

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