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Properties

Label 2-380-95.18-c1-0-6
Degree $2$
Conductor $380$
Sign $0.967 + 0.252i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.13 + 0.656i)5-s + (3.52 − 3.52i)7-s + 3i·9-s + 2.15·11-s + (3.98 − 3.98i)17-s − 4.35i·19-s + (6.35 + 6.35i)23-s + (4.13 − 2.80i)25-s + (−5.22 + 9.85i)35-s + (−1.71 − 1.71i)43-s + (−1.97 − 6.41i)45-s + (−9.67 + 9.67i)47-s − 17.8i·49-s + (−4.59 + 1.41i)55-s − 15.1·61-s + ⋯
L(s)  = 1  + (−0.955 + 0.293i)5-s + (1.33 − 1.33i)7-s + i·9-s + 0.648·11-s + (0.966 − 0.966i)17-s − 0.999i·19-s + (1.32 + 1.32i)23-s + (0.827 − 0.561i)25-s + (−0.882 + 1.66i)35-s + (−0.260 − 0.260i)43-s + (−0.293 − 0.955i)45-s + (−1.41 + 1.41i)47-s − 2.55i·49-s + (−0.619 + 0.190i)55-s − 1.94·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37724 - 0.176898i\)
\(L(\frac12)\) \(\approx\) \(1.37724 - 0.176898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.13 - 0.656i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + (-3.52 + 3.52i)T - 7iT^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-3.98 + 3.98i)T - 17iT^{2} \)
23 \( 1 + (-6.35 - 6.35i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (1.71 + 1.71i)T + 43iT^{2} \)
47 \( 1 + (9.67 - 9.67i)T - 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.1T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-6.91 - 6.91i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (3.64 + 3.64i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−11.16688421107122293110024131250, −10.81496394434706725355417438390, −9.535397301275652991489892422373, −8.247636979553360398691005328187, −7.50250306259323317465462708591, −7.04681907394754923668089677493, −5.10008663725652900977377009438, −4.46512190676424976833232514456, −3.20102127803091220919463051249, −1.23167628337153788174988387330, 1.45032919132638176125782674330, 3.27782969532219733896164387361, 4.44452090643278751919784775730, 5.50389841202927360544604100706, 6.59757012411668078467282844263, 7.981799909641520501324130506822, 8.512109101139119411732299087321, 9.294299888439796461495254966142, 10.68262126881824963982653558136, 11.64333612078868200700512153707

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