Properties

Label 2-380-19.4-c1-0-3
Degree $2$
Conductor $380$
Sign $0.966 - 0.256i$
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  + (0.261 + 1.48i)3-s + (0.766 − 0.642i)5-s + (1.67 − 2.90i)7-s + (0.680 − 0.247i)9-s + (1.23 + 2.14i)11-s + (0.324 − 1.84i)13-s + (1.15 + 0.969i)15-s + (0.341 + 0.124i)17-s + (−3.65 − 2.36i)19-s + (4.75 + 1.72i)21-s + (5.71 + 4.79i)23-s + (0.173 − 0.984i)25-s + (2.80 + 4.86i)27-s + (−3.04 + 1.10i)29-s + (−3.78 + 6.55i)31-s + ⋯
L(s)  = 1  + (0.151 + 0.857i)3-s + (0.342 − 0.287i)5-s + (0.633 − 1.09i)7-s + (0.226 − 0.0825i)9-s + (0.373 + 0.646i)11-s + (0.0900 − 0.510i)13-s + (0.298 + 0.250i)15-s + (0.0829 + 0.0301i)17-s + (−0.839 − 0.543i)19-s + (1.03 + 0.377i)21-s + (1.19 + 1.00i)23-s + (0.0347 − 0.196i)25-s + (0.540 + 0.936i)27-s + (−0.565 + 0.205i)29-s + (−0.679 + 1.17i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64731 + 0.214725i\)
\(L(\frac12)\) \(\approx\) \(1.64731 + 0.214725i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (3.65 + 2.36i)T \)
good3 \( 1 + (-0.261 - 1.48i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-1.67 + 2.90i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.23 - 2.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.324 + 1.84i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.341 - 0.124i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.71 - 4.79i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.04 - 1.10i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.78 - 6.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 + (1.49 + 8.45i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (8.07 - 6.77i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-11.0 + 4.02i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (2.01 + 1.69i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (11.3 + 4.11i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.67 + 7.28i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.31 - 0.840i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (0.0802 - 0.0673i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.34 + 7.63i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.948 - 5.37i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.462 + 0.800i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.38 - 7.86i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (6.64 + 2.41i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.96507370214139674948934950077, −10.58373684611796748095657197191, −9.563778396733456105213448095912, −8.893261121788960879978500356981, −7.60093811459637835940664079922, −6.79547352984897381194925316264, −5.20858811161440117172657343345, −4.46808907468825193822655886950, −3.46390247563468659687574981667, −1.47465051502578330026075542080, 1.62071034802330350571774624376, 2.66134902135247703153820156007, 4.38626079108422046275636511663, 5.75650891320445862188645956377, 6.50992615212335252155735546548, 7.58840963809591501512221636131, 8.537953965494021523284550564396, 9.247348542255357801314618302376, 10.55490123438056780655473474398, 11.43025789489781580228945757085

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