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Properties

Label 2-380-76.75-c1-0-3
Degree $2$
Conductor $380$
Sign $-0.638 - 0.769i$
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  + (−0.976 + 1.02i)2-s − 0.502·3-s + (−0.0929 − 1.99i)4-s − 5-s + (0.490 − 0.513i)6-s − 3.57i·7-s + (2.13 + 1.85i)8-s − 2.74·9-s + (0.976 − 1.02i)10-s + 3.89i·11-s + (0.0466 + 1.00i)12-s + 7.12i·13-s + (3.65 + 3.49i)14-s + 0.502·15-s + (−3.98 + 0.371i)16-s + 6.79·17-s + ⋯
L(s)  = 1  + (−0.690 + 0.723i)2-s − 0.289·3-s + (−0.0464 − 0.998i)4-s − 0.447·5-s + (0.200 − 0.209i)6-s − 1.35i·7-s + (0.754 + 0.656i)8-s − 0.915·9-s + (0.308 − 0.323i)10-s + 1.17i·11-s + (0.0134 + 0.289i)12-s + 1.97i·13-s + (0.978 + 0.933i)14-s + 0.129·15-s + (−0.995 + 0.0928i)16-s + 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.638 - 0.769i$
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: \((2,\ 380,\ (\ :1/2),\ -0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.215921 + 0.459484i\)
\(L(\frac12)\) \(\approx\) \(0.215921 + 0.459484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.976 - 1.02i)T \)
5 \( 1 + T \)
19 \( 1 + (3.48 - 2.62i)T \)
good3 \( 1 + 0.502T + 3T^{2} \)
7 \( 1 + 3.57iT - 7T^{2} \)
11 \( 1 - 3.89iT - 11T^{2} \)
13 \( 1 - 7.12iT - 13T^{2} \)
17 \( 1 - 6.79T + 17T^{2} \)
23 \( 1 - 3.22iT - 23T^{2} \)
29 \( 1 - 2.89iT - 29T^{2} \)
31 \( 1 + 6.29T + 31T^{2} \)
37 \( 1 - 1.99iT - 37T^{2} \)
41 \( 1 + 1.95iT - 41T^{2} \)
43 \( 1 - 1.00iT - 43T^{2} \)
47 \( 1 - 5.80iT - 47T^{2} \)
53 \( 1 + 1.75iT - 53T^{2} \)
59 \( 1 - 4.07T + 59T^{2} \)
61 \( 1 + 7.38T + 61T^{2} \)
67 \( 1 + 6.64T + 67T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 - 6.38T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 3.02iT - 83T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 + 9.60iT - 97T^{2} \)
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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

−11.47083464135994254103475025448, −10.63275537790207151504929808440, −9.797549724020574706984676520029, −8.918430653844273151708006479836, −7.70136103050838378000173572781, −7.18332238749300225329335604651, −6.21487332893646025327599077763, −4.91680750776007166330198167909, −3.90442554696015693525035392514, −1.56603608753742368362536878205, 0.45340584064667714730336137615, 2.69787677844831669880876610788, 3.37556296785727268655196782020, 5.30722710188882550168214342562, 6.02458734414871558356826947838, 7.76264418120914095058542554207, 8.404363014902985983944335477445, 9.026403206891940989205170153151, 10.35307269656357875472372211279, 10.98990056261462101018405980889

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