Properties

Label 2-380-19.18-c2-0-9
Degree $2$
Conductor $380$
Sign $0.568 + 0.822i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.42i·3-s + 2.23·5-s + 9.18·7-s − 2.73·9-s + 11.4·11-s + 10.1i·13-s − 7.65i·15-s − 4.19·17-s + (10.8 + 15.6i)19-s − 31.4i·21-s − 0.952·23-s + 5.00·25-s − 21.4i·27-s + 9.43i·29-s − 8.52i·31-s + ⋯
L(s)  = 1  − 1.14i·3-s + 0.447·5-s + 1.31·7-s − 0.303·9-s + 1.03·11-s + 0.778i·13-s − 0.510i·15-s − 0.246·17-s + (0.568 + 0.822i)19-s − 1.49i·21-s − 0.0414·23-s + 0.200·25-s − 0.795i·27-s + 0.325i·29-s − 0.275i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.568 + 0.822i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.99534 - 1.04584i\)
\(L(\frac12)\) \(\approx\) \(1.99534 - 1.04584i\)
\(L(2)\) 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 \( 1 - 2.23T \)
19 \( 1 + (-10.8 - 15.6i)T \)
good3 \( 1 + 3.42iT - 9T^{2} \)
7 \( 1 - 9.18T + 49T^{2} \)
11 \( 1 - 11.4T + 121T^{2} \)
13 \( 1 - 10.1iT - 169T^{2} \)
17 \( 1 + 4.19T + 289T^{2} \)
23 \( 1 + 0.952T + 529T^{2} \)
29 \( 1 - 9.43iT - 841T^{2} \)
31 \( 1 + 8.52iT - 961T^{2} \)
37 \( 1 + 20.0iT - 1.36e3T^{2} \)
41 \( 1 + 61.2iT - 1.68e3T^{2} \)
43 \( 1 + 57.9T + 1.84e3T^{2} \)
47 \( 1 + 35.3T + 2.20e3T^{2} \)
53 \( 1 + 57.6iT - 2.80e3T^{2} \)
59 \( 1 - 16.7iT - 3.48e3T^{2} \)
61 \( 1 + 32.2T + 3.72e3T^{2} \)
67 \( 1 - 6.07iT - 4.48e3T^{2} \)
71 \( 1 - 113. iT - 5.04e3T^{2} \)
73 \( 1 + 27.9T + 5.32e3T^{2} \)
79 \( 1 - 65.7iT - 6.24e3T^{2} \)
83 \( 1 - 60.2T + 6.88e3T^{2} \)
89 \( 1 - 97.9iT - 7.92e3T^{2} \)
97 \( 1 + 92.0iT - 9.40e3T^{2} \)
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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

−11.34438072755942139930913032755, −10.07978184018036532917423700165, −8.983776965400706996700528507235, −8.113527088474725409247893008175, −7.17811092407041770787519934158, −6.41158139947341331989536589483, −5.23696871861822865858960595894, −3.98118324734573304020713428527, −2.01969716935605406204912394824, −1.33061807840619958553860646237, 1.47023778932234442155924028788, 3.21741732459523443826975944264, 4.54494673693549372706344999748, 5.07648425915519760659387400698, 6.37712499513736499786913039358, 7.68151784510674574549777546333, 8.723111271141943842962899048198, 9.515235342740355011238686159306, 10.34118132072479522652910343779, 11.18567621926066993546194928624

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