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Properties

Label 2-380-380.379-c1-0-12
Degree $2$
Conductor $380$
Sign $0.629 - 0.777i$
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  + (−1.27 + 0.620i)2-s − 0.701i·3-s + (1.22 − 1.57i)4-s + (−1.10 + 1.94i)5-s + (0.435 + 0.891i)6-s + 1.18·7-s + (−0.581 + 2.76i)8-s + 2.50·9-s + (0.196 − 3.15i)10-s − 1.72i·11-s + (−1.10 − 0.862i)12-s − 2.64·13-s + (−1.50 + 0.736i)14-s + (1.36 + 0.775i)15-s + (−0.980 − 3.87i)16-s + 4.62i·17-s + ⋯
L(s)  = 1  + (−0.898 + 0.439i)2-s − 0.405i·3-s + (0.614 − 0.789i)4-s + (−0.493 + 0.869i)5-s + (0.177 + 0.364i)6-s + 0.448·7-s + (−0.205 + 0.978i)8-s + 0.835·9-s + (0.0619 − 0.998i)10-s − 0.518i·11-s + (−0.319 − 0.249i)12-s − 0.733·13-s + (−0.402 + 0.196i)14-s + (0.352 + 0.200i)15-s + (−0.245 − 0.969i)16-s + 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.629 - 0.777i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.629 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.814432 + 0.388561i\)
\(L(\frac12)\) \(\approx\) \(0.814432 + 0.388561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.620i)T \)
5 \( 1 + (1.10 - 1.94i)T \)
19 \( 1 + (-2.08 - 3.82i)T \)
good3 \( 1 + 0.701iT - 3T^{2} \)
7 \( 1 - 1.18T + 7T^{2} \)
11 \( 1 + 1.72iT - 11T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 - 4.62iT - 17T^{2} \)
23 \( 1 - 6.62T + 23T^{2} \)
29 \( 1 - 7.53iT - 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 - 8.43T + 37T^{2} \)
41 \( 1 - 7.40iT - 41T^{2} \)
43 \( 1 + 9.29T + 43T^{2} \)
47 \( 1 - 4.10T + 47T^{2} \)
53 \( 1 + 1.98T + 53T^{2} \)
59 \( 1 - 3.13T + 59T^{2} \)
61 \( 1 - 3.75T + 61T^{2} \)
67 \( 1 + 9.50iT - 67T^{2} \)
71 \( 1 + 9.11T + 71T^{2} \)
73 \( 1 + 6.47iT - 73T^{2} \)
79 \( 1 + 6.38T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + 14.1iT - 89T^{2} \)
97 \( 1 - 3.39T + 97T^{2} \)
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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

−11.29660221109586271126052590256, −10.44261296958577560442787697524, −9.785009614416258683052802028664, −8.403361308364807316377804132428, −7.77950930832345151782105890523, −6.93738375988909227845766427332, −6.16772085931582371557812098308, −4.70987332495527258768533121581, −3.00086942341371077804011024431, −1.40571900932971327555837889964, 0.959135189765981645173436727083, 2.66754486039166606281013384955, 4.29293161233887367224986533807, 4.97047305828753761578969800027, 6.96096038518329413838181880138, 7.60517094744596868829085918864, 8.619746453534096049537284810866, 9.563205739335947044577408910724, 9.996536444867624758577629019370, 11.36128629913429185206490939193

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