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Properties

Label 2-380-76.27-c1-0-28
Degree $2$
Conductor $380$
Sign $0.901 + 0.433i$
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  + (−0.0867 + 1.41i)2-s + (0.980 − 1.69i)3-s + (−1.98 − 0.244i)4-s + (−0.5 + 0.866i)5-s + (2.31 + 1.53i)6-s − 1.97i·7-s + (0.518 − 2.78i)8-s + (−0.421 − 0.730i)9-s + (−1.17 − 0.780i)10-s − 5.82i·11-s + (−2.36 + 3.12i)12-s + (4.54 − 2.62i)13-s + (2.78 + 0.171i)14-s + (0.980 + 1.69i)15-s + (3.87 + 0.972i)16-s + (−2.29 + 3.97i)17-s + ⋯
L(s)  = 1  + (−0.0613 + 0.998i)2-s + (0.565 − 0.980i)3-s + (−0.992 − 0.122i)4-s + (−0.223 + 0.387i)5-s + (0.943 + 0.624i)6-s − 0.745i·7-s + (0.183 − 0.983i)8-s + (−0.140 − 0.243i)9-s + (−0.372 − 0.246i)10-s − 1.75i·11-s + (−0.681 + 0.903i)12-s + (1.26 − 0.727i)13-s + (0.743 + 0.0457i)14-s + (0.253 + 0.438i)15-s + (0.969 + 0.243i)16-s + (−0.556 + 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.901 + 0.433i$
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: \((2,\ 380,\ (\ :1/2),\ 0.901 + 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32379 - 0.301984i\)
\(L(\frac12)\) \(\approx\) \(1.32379 - 0.301984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0867 - 1.41i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (4.33 - 0.434i)T \)
good3 \( 1 + (-0.980 + 1.69i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.97iT - 7T^{2} \)
11 \( 1 + 5.82iT - 11T^{2} \)
13 \( 1 + (-4.54 + 2.62i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.29 - 3.97i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.54 + 1.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.95 + 2.85i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + 6.81iT - 37T^{2} \)
41 \( 1 + (-2.48 - 1.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.44 - 2.56i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (11.5 - 6.66i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.3 - 6.55i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.17 - 8.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.62 + 6.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.99 - 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.0329 + 0.0570i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.29 - 5.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.55 + 2.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.28iT - 83T^{2} \)
89 \( 1 + (-10.0 + 5.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.927 - 0.535i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.96911682259430250366089069240, −10.56401636210349719519531361211, −8.867877944982327316977866931241, −8.243032538217320080617403879631, −7.71310828800350267440053100984, −6.43788582846925799653683322513, −6.09344108972352258653985238578, −4.27472905335857521524853118103, −3.15956085206110703508479237605, −0.976761568739542726529814755353, 1.89579129703923928506843523224, 3.22113265916010099444485541581, 4.43198576021285730705304010142, 4.85537149675535614283995353104, 6.67284117914372097642023375517, 8.273382681856332878007292064987, 9.039507697869492754888251214432, 9.492885183936663212840334653217, 10.41187978899430721187916739449, 11.39296825818081452903574693589

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