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Properties

Label 2-380-20.7-c1-0-34
Degree $2$
Conductor $380$
Sign $0.935 - 0.353i$
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  + (0.00522 + 1.41i)2-s + (−0.212 − 0.212i)3-s + (−1.99 + 0.0147i)4-s + (2.19 + 0.442i)5-s + (0.299 − 0.301i)6-s + (2.65 − 2.65i)7-s + (−0.0313 − 2.82i)8-s − 2.90i·9-s + (−0.614 + 3.10i)10-s − 2.04i·11-s + (0.428 + 0.422i)12-s + (−0.226 + 0.226i)13-s + (3.76 + 3.73i)14-s + (−0.371 − 0.560i)15-s + (3.99 − 0.0590i)16-s + (−3.06 − 3.06i)17-s + ⋯
L(s)  = 1  + (0.00369 + 0.999i)2-s + (−0.122 − 0.122i)3-s + (−0.999 + 0.00738i)4-s + (0.980 + 0.198i)5-s + (0.122 − 0.123i)6-s + (1.00 − 1.00i)7-s + (−0.0110 − 0.999i)8-s − 0.969i·9-s + (−0.194 + 0.980i)10-s − 0.616i·11-s + (0.123 + 0.121i)12-s + (−0.0627 + 0.0627i)13-s + (1.00 + 0.998i)14-s + (−0.0960 − 0.144i)15-s + (0.999 − 0.0147i)16-s + (−0.742 − 0.742i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.935 - 0.353i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.935 - 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44869 + 0.264787i\)
\(L(\frac12)\) \(\approx\) \(1.44869 + 0.264787i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.00522 - 1.41i)T \)
5 \( 1 + (-2.19 - 0.442i)T \)
19 \( 1 + T \)
good3 \( 1 + (0.212 + 0.212i)T + 3iT^{2} \)
7 \( 1 + (-2.65 + 2.65i)T - 7iT^{2} \)
11 \( 1 + 2.04iT - 11T^{2} \)
13 \( 1 + (0.226 - 0.226i)T - 13iT^{2} \)
17 \( 1 + (3.06 + 3.06i)T + 17iT^{2} \)
23 \( 1 + (-5.40 - 5.40i)T + 23iT^{2} \)
29 \( 1 - 6.50iT - 29T^{2} \)
31 \( 1 - 5.78iT - 31T^{2} \)
37 \( 1 + (6.86 + 6.86i)T + 37iT^{2} \)
41 \( 1 - 6.57T + 41T^{2} \)
43 \( 1 + (3.72 + 3.72i)T + 43iT^{2} \)
47 \( 1 + (-1.24 + 1.24i)T - 47iT^{2} \)
53 \( 1 + (6.74 - 6.74i)T - 53iT^{2} \)
59 \( 1 - 4.42T + 59T^{2} \)
61 \( 1 - 0.777T + 61T^{2} \)
67 \( 1 + (5.08 - 5.08i)T - 67iT^{2} \)
71 \( 1 - 1.53iT - 71T^{2} \)
73 \( 1 + (7.28 - 7.28i)T - 73iT^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 + (-4.29 - 4.29i)T + 83iT^{2} \)
89 \( 1 + 8.74iT - 89T^{2} \)
97 \( 1 + (2.60 + 2.60i)T + 97iT^{2} \)
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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

−11.20953041303300564211823801398, −10.45635574996576975006157986980, −9.252705224087780417103677095217, −8.745544674186159893174900023046, −7.28491460328674645897124053255, −6.85060995064821828608783670052, −5.66332514922330071375487513019, −4.81446837882916299210688668557, −3.46107214866014154328518810539, −1.17938467907745452505685139845, 1.84136418841460399033346053627, 2.48211652558656640106902862573, 4.57208243150057167315974954831, 5.07238422375536720288456803354, 6.20098852138631547169874387309, 8.015398437136431482656754143422, 8.709590443927153530972489336924, 9.612009327974462432388719268479, 10.54263034831409201220307700481, 11.17222595252699647976376611743

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