Properties

Label 2-380-95.37-c1-0-2
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: $\chi_{380} (37, \cdot )$
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

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

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