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Properties

Label 2-3800-760.99-c0-0-0
Degree $2$
Conductor $3800$
Sign $0.779 + 0.626i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.779 + 0.626i$
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: \((2,\ 3800,\ (\ :0),\ 0.779 + 0.626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5935739531\)
\(L(\frac12)\) \(\approx\) \(0.5935739531\)
\(L(1)\) 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

−8.901113787554589020123753822949, −7.87113216080578245728752722616, −6.67579645722138511426426769328, −6.12574402520474266006163036162, −5.58031559897304795894436128648, −4.58397158790654365499849287908, −4.37220743375598972766409215075, −3.24118834792484319112290331844, −1.91323602548740400020412953216, −0.78598950242357239580501925649, 0.59516189190521390542563173250, 2.18439565928928402818712760306, 3.81403383829402909260491181492, 4.53212397580944356946099465971, 5.25898141692672780032503732565, 5.75277749803782949682201671207, 6.49411107694637236246842180558, 7.09896268396477670694328100812, 7.57426467832171181672200299551, 8.582040678520319802131286734954

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