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Properties

Label 2-3800-760.99-c0-0-4
Degree $2$
Conductor $3800$
Sign $0.968 - 0.247i$
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  + (−0.342 + 0.939i)2-s + (1.85 − 0.326i)3-s + (−0.766 − 0.642i)4-s + (−0.326 + 1.85i)6-s + (0.866 − 0.500i)8-s + (2.37 − 0.866i)9-s + (−0.173 − 0.300i)11-s + (−1.62 − 0.939i)12-s + (0.173 + 0.984i)16-s + (0.342 − 0.939i)17-s + 2.53i·18-s + (−0.766 − 0.642i)19-s + (0.342 − 0.0603i)22-s + (1.43 − 1.20i)24-s + (2.49 − 1.43i)27-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (1.85 − 0.326i)3-s + (−0.766 − 0.642i)4-s + (−0.326 + 1.85i)6-s + (0.866 − 0.500i)8-s + (2.37 − 0.866i)9-s + (−0.173 − 0.300i)11-s + (−1.62 − 0.939i)12-s + (0.173 + 0.984i)16-s + (0.342 − 0.939i)17-s + 2.53i·18-s + (−0.766 − 0.642i)19-s + (0.342 − 0.0603i)22-s + (1.43 − 1.20i)24-s + (2.49 − 1.43i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.968 - 0.247i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.968 - 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.978947632\)
\(L(\frac12)\) \(\approx\) \(1.978947632\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 \)
19 \( 1 + (0.766 + 0.642i)T \)
good3 \( 1 + (-1.85 + 0.326i)T + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.342 + 0.939i)T + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.524 - 1.43i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.342 - 0.0603i)T + (0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (1.32 + 0.766i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \)
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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

−8.558206995950036825862596866380, −8.088240513151775177581597298115, −7.30253308513532999635934539547, −6.91363573453943233195306153951, −5.96425909772910108439612971634, −4.84388947537812873614976365367, −4.12894536186700440504082747636, −3.16379293037235338019377274810, −2.33062541064871430715829669528, −1.14511383369610404176744166863, 1.59178102936174822211868230152, 2.20997556632584974323114498677, 3.08258648960670453787087711561, 3.84581428758202027805703207547, 4.28658848595000816422418061576, 5.35485000497104824418389638727, 6.77015208130221820749738476263, 7.67815569609872465856616067312, 8.166213544663935690977953543151, 8.696934415470166124296111867641

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