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Properties

Label 2-3800-152.139-c0-0-4
Degree $2$
Conductor $3800$
Sign $0.631 + 0.775i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.300 − 0.173i)7-s + (−0.866 + 0.500i)8-s + (−0.766 − 0.642i)9-s + (−0.766 − 1.32i)11-s + (−0.342 − 0.939i)13-s + (−0.0603 − 0.342i)14-s + (−0.939 − 0.342i)16-s − i·18-s + (−0.766 + 0.642i)19-s + (0.524 − 1.43i)22-s + (1.85 + 0.326i)23-s + (0.5 − 0.866i)26-s + (0.223 − 0.266i)28-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.300 − 0.173i)7-s + (−0.866 + 0.500i)8-s + (−0.766 − 0.642i)9-s + (−0.766 − 1.32i)11-s + (−0.342 − 0.939i)13-s + (−0.0603 − 0.342i)14-s + (−0.939 − 0.342i)16-s − i·18-s + (−0.766 + 0.642i)19-s + (0.524 − 1.43i)22-s + (1.85 + 0.326i)23-s + (0.5 − 0.866i)26-s + (0.223 − 0.266i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.631 + 0.775i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.631 + 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9377856044\)
\(L(\frac12)\) \(\approx\) \(0.9377856044\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 \)
19 \( 1 + (0.766 - 0.642i)T \)
good3 \( 1 + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.300 + 0.173i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-1.85 - 0.326i)T + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.87iT - T^{2} \)
41 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (0.342 + 0.0603i)T + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.173 - 0.984i)T^{2} \)
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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

−8.582147509076677513358079958195, −7.78178078692251458932098166135, −7.00190507726258487576797954224, −6.27023452065248572059846487575, −5.46224495886188465461611485333, −5.22519275327084068249784743296, −3.79298212839026636309648412943, −3.30388673604034837825825341271, −2.54552651539510379931434189431, −0.40174486317415070186052478663, 1.60945670089102455026612486123, 2.57739145307573726846400788545, 3.01065276980924183137763759822, 4.49312287776269092158751599470, 4.75206403663292254351588920914, 5.52131506483257804116002297064, 6.61381911442416819447268218600, 7.02104523859212297588926483856, 8.178732855035040725091151812833, 8.967879113576591457841747486419

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