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Properties

Label 2-380-95.49-c1-0-3
Degree $2$
Conductor $380$
Sign $0.574 - 0.818i$
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  + (1.74 + 1.00i)3-s + (0.0408 + 2.23i)5-s − 1.34i·7-s + (0.526 + 0.912i)9-s + 5.25·11-s + (−2.10 + 1.21i)13-s + (−2.17 + 3.93i)15-s + (1.17 + 0.679i)17-s + (2.89 + 3.25i)19-s + (1.35 − 2.34i)21-s + (−7.05 + 4.07i)23-s + (−4.99 + 0.182i)25-s − 3.91i·27-s + (−1.03 − 1.79i)29-s − 0.513·31-s + ⋯
L(s)  = 1  + (1.00 + 0.581i)3-s + (0.0182 + 0.999i)5-s − 0.507i·7-s + (0.175 + 0.304i)9-s + 1.58·11-s + (−0.584 + 0.337i)13-s + (−0.562 + 1.01i)15-s + (0.285 + 0.164i)17-s + (0.664 + 0.746i)19-s + (0.295 − 0.511i)21-s + (−1.47 + 0.849i)23-s + (−0.999 + 0.0365i)25-s − 0.754i·27-s + (−0.192 − 0.333i)29-s − 0.0921·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.574 - 0.818i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.574 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68207 + 0.874053i\)
\(L(\frac12)\) \(\approx\) \(1.68207 + 0.874053i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0408 - 2.23i)T \)
19 \( 1 + (-2.89 - 3.25i)T \)
good3 \( 1 + (-1.74 - 1.00i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 1.34iT - 7T^{2} \)
11 \( 1 - 5.25T + 11T^{2} \)
13 \( 1 + (2.10 - 1.21i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.17 - 0.679i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (7.05 - 4.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.03 + 1.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.513T + 31T^{2} \)
37 \( 1 + 5.57iT - 37T^{2} \)
41 \( 1 + (-2.70 + 4.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (11.0 + 6.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.82 + 1.63i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.1 + 5.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0175 + 0.0304i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.518 - 0.897i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.664 + 0.383i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.68 + 9.84i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.86 + 1.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.48 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.20iT - 83T^{2} \)
89 \( 1 + (3.65 + 6.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.721 - 0.416i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.56744638222884979817657292054, −10.21816321134354162082571450801, −9.780310399752036250004066459477, −8.887323089996859075869025406721, −7.74540754807614489866083193287, −6.92724519588910304116003140350, −5.81033969968730870409708126402, −3.88644479274332996043337154379, −3.66951061486645135449630445292, −2.07212380285677557416276171305, 1.39904924003395641531675599451, 2.71301223118659166404111143894, 4.11941726757174274441908858717, 5.32880105059124455595004827104, 6.57020639400897560512110829158, 7.70300639819021440179626666950, 8.514965984842025364853789277724, 9.153263336977056690925548537231, 9.935702399010335634185902982145, 11.61699976546072852727225602338

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