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Properties

Label 2-3800-760.189-c0-0-14
Degree $2$
Conductor $3800$
Sign $0.447 + 0.894i$
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  + i·2-s − 0.347i·3-s − 4-s + 0.347·6-s − 1.87i·7-s − i·8-s + 0.879·9-s + 0.347i·12-s − 1.53i·13-s + 1.87·14-s + 16-s + 1.53i·17-s + 0.879i·18-s − 19-s − 0.652·21-s + ⋯
L(s)  = 1  + i·2-s − 0.347i·3-s − 4-s + 0.347·6-s − 1.87i·7-s − i·8-s + 0.879·9-s + 0.347i·12-s − 1.53i·13-s + 1.87·14-s + 16-s + 1.53i·17-s + 0.879i·18-s − 19-s − 0.652·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9694149988\)
\(L(\frac12)\) \(\approx\) \(0.9694149988\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 0.347iT - T^{2} \)
7 \( 1 + 1.87iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.53iT - T^{2} \)
17 \( 1 - 1.53iT - T^{2} \)
23 \( 1 + 0.347iT - T^{2} \)
29 \( 1 + 1.53T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - 1.87iT - T^{2} \)
59 \( 1 + 0.347T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.87iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.347iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.132989071453974875149839510057, −7.70324661072561302618240048991, −7.21606534151767797271862674683, −6.43146584569491643170574918343, −5.80847710748034343931523372180, −4.71758776344464674205871723641, −4.01147978866766937997893607457, −3.53530464658958635503126555797, −1.68525463392386657561590875773, −0.56370632011863495275363234968, 1.70478601775394990787670295736, 2.28381096615094497585937348680, 3.22465770066772047932746935620, 4.23774687609070138180374282642, 4.86512787631152544846727904971, 5.55581318457085695706283637694, 6.50131493694387895517405604411, 7.41047164321565970116125081102, 8.531103414624087194908622046897, 8.973915581017129654469215583479

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