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Properties

Label 2-1900-19.7-c1-0-29
Degree $2$
Conductor $1900$
Sign $-0.550 - 0.834i$
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  + (−1.26 − 2.18i)3-s − 2.72·7-s + (−1.67 + 2.90i)9-s + 3.31·11-s + (1.62 − 2.81i)13-s + (−1.17 − 2.03i)17-s + (−3.11 − 3.04i)19-s + (3.43 + 5.95i)21-s + (1.07 − 1.86i)23-s + 0.893·27-s + (1.96 − 3.40i)29-s − 10.1·31-s + (−4.17 − 7.23i)33-s + 3.68·37-s − 8.18·39-s + ⋯
L(s)  = 1  + (−0.727 − 1.26i)3-s − 1.03·7-s + (−0.559 + 0.968i)9-s + 0.999·11-s + (0.450 − 0.780i)13-s + (−0.285 − 0.494i)17-s + (−0.714 − 0.699i)19-s + (0.750 + 1.29i)21-s + (0.223 − 0.387i)23-s + 0.171·27-s + (0.365 − 0.632i)29-s − 1.83·31-s + (−0.727 − 1.25i)33-s + 0.605·37-s − 1.31·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.550 - 0.834i$
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: \((2,\ 1900,\ (\ :1/2),\ -0.550 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3121415154\)
\(L(\frac12)\) \(\approx\) \(0.3121415154\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.11 + 3.04i)T \)
good3 \( 1 + (1.26 + 2.18i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
13 \( 1 + (-1.62 + 2.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.17 + 2.03i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.07 + 1.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.96 + 3.40i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 3.68T + 37T^{2} \)
41 \( 1 + (-0.363 - 0.629i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.18 + 2.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.51 - 9.54i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.49 - 7.77i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.48 - 9.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.22 + 7.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.87 - 8.44i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.45 + 5.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.24 + 2.14i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.99 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.68T + 83T^{2} \)
89 \( 1 + (4.27 - 7.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.61 - 6.26i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.741977905169460537799382495756, −7.62601706234999598016312227566, −7.02298074395710032655892984737, −6.24058862241112069217876072784, −5.96518849613073451775692842148, −4.69498630468099006617387131055, −3.54726074963370147548749564322, −2.48005680526893523549163968602, −1.20825087222010745546456252904, −0.13568595109921765479081352163, 1.75994741852242021161108994368, 3.52334188023016510025323219090, 3.84349547497553739576467301279, 4.78590148368671162960166929888, 5.78085676321234792732283298230, 6.39171012887909551476009883902, 7.08263149984299130583268150560, 8.523391705008497407964498076622, 9.122063138679836012179534781614, 9.819992037873929747655502804022

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