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Properties

Label 2-3800-152.131-c0-0-5
Degree $2$
Conductor $3800$
Sign $-0.189 + 0.981i$
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 + (−0.266 − 0.223i)3-s + (−0.939 − 0.342i)4-s + (0.266 − 0.223i)6-s + (0.5 − 0.866i)8-s + (−0.152 − 0.866i)9-s + (−0.766 + 1.32i)11-s + (0.173 + 0.300i)12-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + 0.879·18-s + (−0.939 − 0.342i)19-s + (−1.17 − 0.984i)22-s + (−0.326 + 0.118i)24-s + (−0.326 + 0.565i)27-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.266 − 0.223i)3-s + (−0.939 − 0.342i)4-s + (0.266 − 0.223i)6-s + (0.5 − 0.866i)8-s + (−0.152 − 0.866i)9-s + (−0.766 + 1.32i)11-s + (0.173 + 0.300i)12-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + 0.879·18-s + (−0.939 − 0.342i)19-s + (−1.17 − 0.984i)22-s + (−0.326 + 0.118i)24-s + (−0.326 + 0.565i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2409417154\)
\(L(\frac12)\) \(\approx\) \(0.2409417154\)
\(L(1)\) 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 \( 1 + (0.939 + 0.342i)T \)
good3 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
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 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
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

−8.430288526080192764186721374742, −7.52042483320470657897726672305, −6.99469118156736561098968252179, −6.45588296885371684929170856879, −5.52060509807175794276420049309, −4.89030433668280668522632912268, −4.14867046600230301750422647157, −3.01600727090980773279987997249, −1.69981268090546034629691310302, −0.14961762086333696360935269126, 1.51538650696186285296774844298, 2.50540148636659873801102332292, 3.36067603196106313675791407586, 4.16672468280027395878170425172, 5.09920629513079790642830955535, 5.63405738128980571205764579543, 6.55353230944259758682397581768, 7.88512327069241846971942814854, 8.270103621624143949922350970897, 8.755768467589429968913737265955

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