Properties

Label 2-380-380.219-c1-0-9
Degree $2$
Conductor $380$
Sign $0.685 + 0.728i$
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  + (−0.777 − 1.18i)2-s + (−1.85 − 2.20i)3-s + (−0.790 + 1.83i)4-s + (−0.543 + 2.16i)5-s + (−1.16 + 3.90i)6-s + (0.0958 − 0.165i)7-s + (2.78 − 0.494i)8-s + (−0.920 + 5.22i)9-s + (2.98 − 1.04i)10-s + (0.219 − 0.126i)11-s + (5.51 − 1.65i)12-s + (1.99 + 1.67i)13-s + (−0.270 + 0.0158i)14-s + (5.79 − 2.81i)15-s + (−2.74 − 2.90i)16-s + (2.07 − 0.365i)17-s + ⋯
L(s)  = 1  + (−0.549 − 0.835i)2-s + (−1.06 − 1.27i)3-s + (−0.395 + 0.918i)4-s + (−0.243 + 0.970i)5-s + (−0.476 + 1.59i)6-s + (0.0362 − 0.0627i)7-s + (0.984 − 0.174i)8-s + (−0.306 + 1.74i)9-s + (0.943 − 0.330i)10-s + (0.0663 − 0.0382i)11-s + (1.59 − 0.478i)12-s + (0.553 + 0.464i)13-s + (−0.0723 + 0.00423i)14-s + (1.49 − 0.727i)15-s + (−0.687 − 0.726i)16-s + (0.503 − 0.0887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.685 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584187 - 0.252450i\)
\(L(\frac12)\) \(\approx\) \(0.584187 - 0.252450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.777 + 1.18i)T \)
5 \( 1 + (0.543 - 2.16i)T \)
19 \( 1 + (-4.07 - 1.54i)T \)
good3 \( 1 + (1.85 + 2.20i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (-0.0958 + 0.165i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.219 + 0.126i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.99 - 1.67i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.07 + 0.365i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (4.63 - 1.68i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-6.52 - 1.15i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.47 + 2.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.57T + 37T^{2} \)
41 \( 1 + (7.06 + 8.41i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-6.67 - 2.42i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.936 - 5.31i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-4.80 + 1.74i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.04 - 11.5i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-11.1 + 4.06i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.04 - 0.360i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-9.44 - 3.43i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-2.30 - 2.74i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (5.82 - 4.88i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.59 + 2.76i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.40 - 10.0i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-2.46 - 13.9i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.42541788545950346246600749163, −10.60283924690835507320335028771, −9.721779220374040861291850731480, −8.215520128824704759604511526292, −7.48366309576185210570855914282, −6.69058536459102008567623113663, −5.65072376473483994405277783863, −3.94958441900834832002243768234, −2.50059938103932678324844950542, −1.09771276012533583771834262996, 0.77332672403431180537651591182, 3.92337822805430839137374170161, 4.91692198588860767462155999215, 5.53010896574646217819015676341, 6.47033123687569479913904026604, 7.954334564653848046374285834892, 8.750669399230912524561725287606, 9.785863887942189227803226704873, 10.21398198597158236232281255617, 11.33652846813826276049433332208

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