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Properties

Label 2-380-76.75-c1-0-12
Degree $2$
Conductor $380$
Sign $-0.543 - 0.839i$
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  + (−0.139 + 1.40i)2-s + 1.11·3-s + (−1.96 − 0.391i)4-s + 5-s + (−0.154 + 1.56i)6-s + 3.02i·7-s + (0.824 − 2.70i)8-s − 1.76·9-s + (−0.139 + 1.40i)10-s + 3.96i·11-s + (−2.18 − 0.435i)12-s + 0.110i·13-s + (−4.25 − 0.421i)14-s + 1.11·15-s + (3.69 + 1.53i)16-s + 2.37·17-s + ⋯
L(s)  = 1  + (−0.0984 + 0.995i)2-s + 0.642·3-s + (−0.980 − 0.195i)4-s + 0.447·5-s + (−0.0632 + 0.639i)6-s + 1.14i·7-s + (0.291 − 0.956i)8-s − 0.587·9-s + (−0.0440 + 0.445i)10-s + 1.19i·11-s + (−0.629 − 0.125i)12-s + 0.0305i·13-s + (−1.13 − 0.112i)14-s + 0.287·15-s + (0.923 + 0.384i)16-s + 0.574·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.543 - 0.839i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.543 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678320 + 1.24708i\)
\(L(\frac12)\) \(\approx\) \(0.678320 + 1.24708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.139 - 1.40i)T \)
5 \( 1 - T \)
19 \( 1 + (-3.03 - 3.12i)T \)
good3 \( 1 - 1.11T + 3T^{2} \)
7 \( 1 - 3.02iT - 7T^{2} \)
11 \( 1 - 3.96iT - 11T^{2} \)
13 \( 1 - 0.110iT - 13T^{2} \)
17 \( 1 - 2.37T + 17T^{2} \)
23 \( 1 - 1.44iT - 23T^{2} \)
29 \( 1 - 1.94iT - 29T^{2} \)
31 \( 1 - 0.435T + 31T^{2} \)
37 \( 1 + 4.77iT - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + 8.64iT - 43T^{2} \)
47 \( 1 - 0.861iT - 47T^{2} \)
53 \( 1 + 2.99iT - 53T^{2} \)
59 \( 1 - 15.1T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 9.40T + 67T^{2} \)
71 \( 1 - 6.06T + 71T^{2} \)
73 \( 1 - 9.61T + 73T^{2} \)
79 \( 1 - 8.29T + 79T^{2} \)
83 \( 1 - 4.17iT - 83T^{2} \)
89 \( 1 + 1.53iT - 89T^{2} \)
97 \( 1 - 7.56iT - 97T^{2} \)
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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

−11.98154988215608099635457158992, −10.35966315282497660748637054998, −9.426893211268069837149900417148, −8.922783347090223167974858209312, −7.965488922896898274280418044371, −7.07299163635200007757258530015, −5.76967776160351338733743009951, −5.24246663846569090248375756935, −3.64390414445879280134354672853, −2.13005496437079041561960577384, 0.990658244188030031351196935459, 2.77228497441989343233610012890, 3.53777856910440716200334378822, 4.85714369073065246831409681980, 6.11891588748698812159497322106, 7.67960468480006605669960629032, 8.428828325641818687286509693777, 9.366554210913411615476434437276, 10.13509330601142430272686130821, 11.07550635079353617897484363189

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