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Properties

Label 2-3800-152.123-c0-0-5
Degree $2$
Conductor $3800$
Sign $0.944 + 0.327i$
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  + (0.173 + 0.984i)2-s + (1.17 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (1.17 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.233 − 1.32i)9-s + (−0.173 − 0.300i)11-s + (−0.766 + 1.32i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 1.34·18-s + (0.173 − 0.984i)19-s + (0.266 − 0.223i)22-s + (−1.43 − 0.524i)24-s + (−0.266 − 0.460i)27-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (1.17 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (1.17 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.233 − 1.32i)9-s + (−0.173 − 0.300i)11-s + (−0.766 + 1.32i)12-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s + 1.34·18-s + (0.173 − 0.984i)19-s + (0.266 − 0.223i)22-s + (−1.43 − 0.524i)24-s + (−0.266 − 0.460i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (2251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.806412501\)
\(L(\frac12)\) \(\approx\) \(1.806412501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 \)
19 \( 1 + (-0.173 + 0.984i)T \)
good3 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.478657764368099879577835276343, −7.77092446414334104322083019158, −7.23581077030016829258891000146, −6.73967745355069582831756420916, −5.81853540938172331984199403967, −4.97417426534459312754403395744, −4.03256619496704889360884073084, −3.06188747286456195022491006060, −2.40187061466848050513583326748, −0.893618540191142883717432693184, 1.56323275383286706708476674064, 2.50227271882891344922748469175, 3.24985887428549356795438023698, 4.02159505318365359416020440057, 4.46809401019238440818952385069, 5.44809276557153515644871523758, 6.29433553104965002266899280784, 7.74646163338743878916965572972, 8.164826046355360661569260674713, 8.993304511775089781572505851126

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