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Properties

Label 2-380-76.27-c1-0-23
Degree $2$
Conductor $380$
Sign $0.105 + 0.994i$
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  + (−1.10 − 0.880i)2-s + (0.736 − 1.27i)3-s + (0.449 + 1.94i)4-s + (−0.5 + 0.866i)5-s + (−1.93 + 0.763i)6-s + 0.456i·7-s + (1.21 − 2.55i)8-s + (0.414 + 0.717i)9-s + (1.31 − 0.518i)10-s − 4.12i·11-s + (2.81 + 0.861i)12-s + (1.19 − 0.690i)13-s + (0.401 − 0.504i)14-s + (0.736 + 1.27i)15-s + (−3.59 + 1.75i)16-s + (2.62 − 4.54i)17-s + ⋯
L(s)  = 1  + (−0.782 − 0.622i)2-s + (0.425 − 0.736i)3-s + (0.224 + 0.974i)4-s + (−0.223 + 0.387i)5-s + (−0.791 + 0.311i)6-s + 0.172i·7-s + (0.430 − 0.902i)8-s + (0.138 + 0.239i)9-s + (0.416 − 0.163i)10-s − 1.24i·11-s + (0.813 + 0.248i)12-s + (0.331 − 0.191i)13-s + (0.107 − 0.134i)14-s + (0.190 + 0.329i)15-s + (−0.898 + 0.438i)16-s + (0.636 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.105 + 0.994i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.105 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802652 - 0.721744i\)
\(L(\frac12)\) \(\approx\) \(0.802652 - 0.721744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 + 0.880i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-4.15 + 1.30i)T \)
good3 \( 1 + (-0.736 + 1.27i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 0.456iT - 7T^{2} \)
11 \( 1 + 4.12iT - 11T^{2} \)
13 \( 1 + (-1.19 + 0.690i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.62 + 4.54i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.10 + 4.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.58 - 4.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.72T + 31T^{2} \)
37 \( 1 - 3.87iT - 37T^{2} \)
41 \( 1 + (0.344 + 0.198i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.80 + 5.08i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.49 + 4.32i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.85 - 1.64i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.81 - 3.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.29 + 3.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.68 - 9.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.118 - 0.204i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.15 - 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.38 - 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.73iT - 83T^{2} \)
89 \( 1 + (-7.61 + 4.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.52 - 0.877i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.14106584844372829615666674847, −10.35384408731402749499410720450, −9.115039253470455440306570311172, −8.487556133998099258998251880056, −7.44494148257398489355260421648, −6.94126822400509233530281456759, −5.30791516544946557893004204595, −3.45140227649251596225987593764, −2.67002458283925299976171989458, −1.04708365035423092352220399340, 1.53345176143766819242022282600, 3.60205328454096911929185548393, 4.73596816095209373410758160349, 5.83269275372434966143560833534, 7.17253475846636284675800005331, 7.82774881177061410195041773754, 9.061106743559397232787233470900, 9.501357356478176754451557883854, 10.29607850831447994370043916145, 11.27909298579404573771535098991

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