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Properties

Label 2-3800-152.35-c0-0-3
Degree $2$
Conductor $3800$
Sign $0.188 + 0.982i$
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  + (−0.766 − 0.642i)2-s + (1.43 − 0.524i)3-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)6-s + (0.500 − 0.866i)8-s + (1.03 − 0.866i)9-s + (0.939 − 1.62i)11-s + (0.766 + 1.32i)12-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s − 1.34·18-s + (0.173 + 0.984i)19-s + (−1.76 + 0.642i)22-s + (0.266 − 1.50i)24-s + (0.266 − 0.460i)27-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (1.43 − 0.524i)3-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)6-s + (0.500 − 0.866i)8-s + (1.03 − 0.866i)9-s + (0.939 − 1.62i)11-s + (0.766 + 1.32i)12-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s − 1.34·18-s + (0.173 + 0.984i)19-s + (−1.76 + 0.642i)22-s + (0.266 − 1.50i)24-s + (0.266 − 0.460i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.188 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.596863017\)
\(L(\frac12)\) \(\approx\) \(1.596863017\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-0.173 - 0.984i)T \)
good3 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.518728157273354778866535136424, −8.063757410138753036047283010885, −7.46175747900631215452995322388, −6.53885836229998901212250633226, −5.76433482287113747945226134915, −4.15490901312524603297962828187, −3.40855795502559021738878300642, −3.07546935319514519677801288150, −1.86393399172571876064177440809, −1.16414841980705346392993377780, 1.45223696666061835205671100235, 2.35681088816189261826820645353, 3.27763253592235253427778556120, 4.44115732235627118357036754470, 4.85272830430224999361430437319, 6.07855195727046430716903194470, 7.05711739594252258225790290505, 7.42436031717628799647343862578, 8.146668712539112828265315621407, 9.059766421456119454654556736060

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