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Properties

Label 2-380-95.13-c1-0-8
Degree $2$
Conductor $380$
Sign $0.0683 + 0.997i$
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  + (1.80 − 0.839i)3-s + (−1.74 − 1.40i)5-s + (−0.755 − 2.81i)7-s + (0.609 − 0.726i)9-s + (1.17 + 2.03i)11-s + (2.20 − 4.73i)13-s + (−4.31 − 1.05i)15-s + (0.125 − 1.42i)17-s + (−3.05 − 3.11i)19-s + (−3.72 − 4.44i)21-s + (2.21 − 1.55i)23-s + (1.07 + 4.88i)25-s + (−1.05 + 3.93i)27-s + (5.56 + 4.67i)29-s + (2.34 + 1.35i)31-s + ⋯
L(s)  = 1  + (1.03 − 0.484i)3-s + (−0.779 − 0.626i)5-s + (−0.285 − 1.06i)7-s + (0.203 − 0.242i)9-s + (0.353 + 0.612i)11-s + (0.611 − 1.31i)13-s + (−1.11 − 0.273i)15-s + (0.0303 − 0.346i)17-s + (−0.700 − 0.713i)19-s + (−0.813 − 0.969i)21-s + (0.461 − 0.323i)23-s + (0.215 + 0.976i)25-s + (−0.203 + 0.757i)27-s + (1.03 + 0.867i)29-s + (0.421 + 0.243i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0683 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0683 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0683 + 0.997i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.0683 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14858 - 1.07262i\)
\(L(\frac12)\) \(\approx\) \(1.14858 - 1.07262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.74 + 1.40i)T \)
19 \( 1 + (3.05 + 3.11i)T \)
good3 \( 1 + (-1.80 + 0.839i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (0.755 + 2.81i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.17 - 2.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.20 + 4.73i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (-0.125 + 1.42i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (-2.21 + 1.55i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (-5.56 - 4.67i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.34 - 1.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.32 + 3.32i)T + 37iT^{2} \)
41 \( 1 + (1.41 - 3.89i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (3.51 - 5.02i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (-8.34 + 0.729i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (-2.51 - 3.59i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (-7.59 + 6.37i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.68 - 9.54i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.534 + 6.10i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (-4.76 + 0.839i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-5.54 - 11.8i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (16.5 + 6.01i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-13.1 + 3.51i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (4.65 - 1.69i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-5.32 - 0.466i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.05487388032091580594623516593, −10.24598962074985589551343962560, −8.973276232106331052135822107393, −8.354013933961408531484035850897, −7.50041173316848899155652520792, −6.77520663722359950287095530846, −5.02786610342459932723487919701, −3.91504923137888326879173950808, −2.88445986111576703017004843709, −1.01466078414141117525366781802, 2.34754378316500622711121212908, 3.46286562996205097300001699737, 4.20000763319114599835195338935, 5.96728591281231738228515108376, 6.81831839001251196231085046248, 8.329766091994390423547042869349, 8.621526634400852226116760354862, 9.559877627167166415226479049672, 10.61585418959986374989956966288, 11.69547688831991039467369980849

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