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Properties

Label 2-380-76.31-c1-0-16
Degree $2$
Conductor $380$
Sign $-0.0315 - 0.999i$
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  + (1.25 + 0.648i)2-s + (0.665 + 1.15i)3-s + (1.15 + 1.62i)4-s + (−0.5 − 0.866i)5-s + (0.0890 + 1.87i)6-s + 4.68i·7-s + (0.400 + 2.79i)8-s + (0.615 − 1.06i)9-s + (−0.0669 − 1.41i)10-s − 4.21i·11-s + (−1.10 + 2.41i)12-s + (−2.65 − 1.53i)13-s + (−3.03 + 5.89i)14-s + (0.665 − 1.15i)15-s + (−1.31 + 3.77i)16-s + (0.913 + 1.58i)17-s + ⋯
L(s)  = 1  + (0.888 + 0.458i)2-s + (0.384 + 0.665i)3-s + (0.579 + 0.814i)4-s + (−0.223 − 0.387i)5-s + (0.0363 + 0.767i)6-s + 1.77i·7-s + (0.141 + 0.989i)8-s + (0.205 − 0.355i)9-s + (−0.0211 − 0.446i)10-s − 1.27i·11-s + (−0.319 + 0.698i)12-s + (−0.737 − 0.425i)13-s + (−0.812 + 1.57i)14-s + (0.171 − 0.297i)15-s + (−0.328 + 0.944i)16-s + (0.221 + 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0315 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0315 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.0315 - 0.999i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.0315 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68263 + 1.73654i\)
\(L(\frac12)\) \(\approx\) \(1.68263 + 1.73654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.648i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (2.30 + 3.69i)T \)
good3 \( 1 + (-0.665 - 1.15i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 4.68iT - 7T^{2} \)
11 \( 1 + 4.21iT - 11T^{2} \)
13 \( 1 + (2.65 + 1.53i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.913 - 1.58i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-5.23 - 3.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.78 + 1.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.61T + 31T^{2} \)
37 \( 1 + 2.54iT - 37T^{2} \)
41 \( 1 + (-10.9 + 6.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.19 - 4.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.93 + 4.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.880 + 0.508i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.293 - 0.508i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.20 + 7.28i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.92 + 3.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.658 - 1.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.57 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.62 - 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.76iT - 83T^{2} \)
89 \( 1 + (5.73 + 3.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.590 - 0.340i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.77465063469664460228753348313, −10.96297123339330078180401553016, −9.434070129853659027502269428160, −8.763212797783179416450424767368, −8.030854958550372285247030834406, −6.56299010205529767648312746632, −5.58596948546264738140050265763, −4.84206779256987730415256417644, −3.49613858034437642447707448883, −2.62016645660364661304208854003, 1.43101963625344899736151759484, 2.74365089854266195695089721456, 4.17313827627991043755731679882, 4.78853262600945812728652905627, 6.69440328119539506183483053556, 7.12311847297027010970881618539, 7.87579675668985487469482095359, 9.782764698159563843919697140516, 10.31254182621888193140448190430, 11.15967791360874412248181621255

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