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Properties

Label 2-1900-19.7-c1-0-4
Degree $2$
Conductor $1900$
Sign $0.807 - 0.589i$
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  + (−1.00 − 1.74i)3-s − 1.34·7-s + (−0.526 + 0.912i)9-s + 5.25·11-s + (−1.21 + 2.10i)13-s + (0.679 + 1.17i)17-s + (−2.89 + 3.25i)19-s + (1.35 + 2.34i)21-s + (−4.07 + 7.05i)23-s − 3.91·27-s + (1.03 − 1.79i)29-s − 0.513·31-s + (−5.29 − 9.16i)33-s − 5.57·37-s + 4.90·39-s + ⋯
L(s)  = 1  + (−0.581 − 1.00i)3-s − 0.507·7-s + (−0.175 + 0.304i)9-s + 1.58·11-s + (−0.337 + 0.584i)13-s + (0.164 + 0.285i)17-s + (−0.664 + 0.746i)19-s + (0.295 + 0.511i)21-s + (−0.849 + 1.47i)23-s − 0.754·27-s + (0.192 − 0.333i)29-s − 0.0921·31-s + (−0.921 − 1.59i)33-s − 0.915·37-s + 0.785·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.807 - 0.589i$
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: \((2,\ 1900,\ (\ :1/2),\ 0.807 - 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.019668618\)
\(L(\frac12)\) \(\approx\) \(1.019668618\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.89 - 3.25i)T \)
good3 \( 1 + (1.00 + 1.74i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 1.34T + 7T^{2} \)
11 \( 1 - 5.25T + 11T^{2} \)
13 \( 1 + (1.21 - 2.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.679 - 1.17i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.07 - 7.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.03 + 1.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.513T + 31T^{2} \)
37 \( 1 + 5.57T + 37T^{2} \)
41 \( 1 + (-2.70 - 4.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.36 - 11.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.63 - 2.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.88 + 10.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.0175 + 0.0304i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.518 + 0.897i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.383 - 0.664i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.68 - 9.84i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.07 - 1.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.48 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.20T + 83T^{2} \)
89 \( 1 + (-3.65 + 6.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.416 - 0.721i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.564361405141748228842860646677, −8.380525184807376574626732164924, −7.58550191802380331887354943727, −6.69176272072300218116942902935, −6.35372965803227835832960274600, −5.59077503297242740723021173623, −4.22656358437086913402121366244, −3.54958803969096594467863004055, −1.98424126111597573169556672062, −1.18469787726902202159307723812, 0.44523806756299251153532478942, 2.19724750615071494647087033631, 3.51280016520254952529749048842, 4.22326823957494083785178344155, 4.96417154313288465704726756501, 5.92829185353256612897030214680, 6.60709299810380490665042484715, 7.43978123503753385393911495311, 8.702300645021788443919611920080, 9.156454329835802338476751906191

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