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Properties

Label 2-3800-152.43-c0-0-6
Degree $2$
Conductor $3800$
Sign $-0.977 + 0.211i$
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  + (0.939 − 0.342i)2-s + (0.326 − 1.85i)3-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)6-s + (0.500 − 0.866i)8-s + (−2.37 − 0.866i)9-s + (−0.173 + 0.300i)11-s + (−0.939 − 1.62i)12-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s − 2.53·18-s + (0.766 − 0.642i)19-s + (−0.0603 + 0.342i)22-s + (−1.43 − 1.20i)24-s + (−1.43 + 2.49i)27-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.326 − 1.85i)3-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)6-s + (0.500 − 0.866i)8-s + (−2.37 − 0.866i)9-s + (−0.173 + 0.300i)11-s + (−0.939 − 1.62i)12-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s − 2.53·18-s + (0.766 − 0.642i)19-s + (−0.0603 + 0.342i)22-s + (−1.43 − 1.20i)24-s + (−1.43 + 2.49i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.361524688\)
\(L(\frac12)\) \(\approx\) \(2.361524688\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 \)
19 \( 1 + (-0.766 + 0.642i)T \)
good3 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
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 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
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 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
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 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
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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

−8.081702260705988670116485047593, −7.29649997847082572946025794158, −6.92343694105425226278139526008, −6.15197252557857084869122831634, −5.52685003081596277956747308775, −4.53210114993937271293616585513, −3.42024467240444856171877485873, −2.55907002462088296955854447643, −1.97564559282647654538059889524, −0.923238296643777423629087500326, 2.25963335511131837605370604285, 3.14879036712836642016871818604, 3.71442409939332230991417682180, 4.52334889933839095118769147383, 5.06105476927046072927589127071, 5.75290470336370749656748974600, 6.53971710243419706209711592717, 7.66891602442064585234947475336, 8.303337517063285920990131496662, 9.041087880099128916423998756445

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