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Properties

Label 2-380-95.64-c1-0-9
Degree $2$
Conductor $380$
Sign $0.132 + 0.991i$
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  + (2.48 − 1.43i)3-s + (−2.21 + 0.325i)5-s − 3.54i·7-s + (2.62 − 4.54i)9-s − 1.81·11-s + (−2.78 − 1.60i)13-s + (−5.03 + 3.98i)15-s + (6.92 − 3.99i)17-s + (0.863 + 4.27i)19-s + (−5.09 − 8.82i)21-s + (7.30 + 4.21i)23-s + (4.78 − 1.43i)25-s − 6.46i·27-s + (−4.29 + 7.43i)29-s − 1.70·31-s + ⋯
L(s)  = 1  + (1.43 − 0.829i)3-s + (−0.989 + 0.145i)5-s − 1.34i·7-s + (0.875 − 1.51i)9-s − 0.547·11-s + (−0.771 − 0.445i)13-s + (−1.30 + 1.02i)15-s + (1.67 − 0.969i)17-s + (0.198 + 0.980i)19-s + (−1.11 − 1.92i)21-s + (1.52 + 0.878i)23-s + (0.957 − 0.287i)25-s − 1.24i·27-s + (−0.796 + 1.38i)29-s − 0.306·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33366 - 1.16762i\)
\(L(\frac12)\) \(\approx\) \(1.33366 - 1.16762i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.325i)T \)
19 \( 1 + (-0.863 - 4.27i)T \)
good3 \( 1 + (-2.48 + 1.43i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 3.54iT - 7T^{2} \)
11 \( 1 + 1.81T + 11T^{2} \)
13 \( 1 + (2.78 + 1.60i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.92 + 3.99i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.30 - 4.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.29 - 7.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + 5.50iT - 37T^{2} \)
41 \( 1 + (-4.05 - 7.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.35 - 2.51i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.16 + 0.674i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.92 - 1.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.960 - 1.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.83 + 4.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.04 + 4.64i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.94 + 5.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.82 + 1.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.08 - 3.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.30iT - 83T^{2} \)
89 \( 1 + (-2.73 + 4.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.91 - 3.99i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.14864424513546729925540009583, −10.11850165297537039121028488307, −9.202446447927930862170396891946, −7.911422582522224856445076395859, −7.53573254337078677630640131703, −7.11083995531319886180173908266, −5.11041537809232966638626011539, −3.57370815969145425063327372758, −3.06511924896433368291802518274, −1.13175118162727061657257292508, 2.45950699808882402987926574162, 3.28739219444754183040540273317, 4.46566354394546475349161094072, 5.44013256463677356941397793729, 7.25209957053765855921485601722, 8.141169094792082775792926572911, 8.772614873606424728057929824263, 9.496785766824658770909686095283, 10.45241892095192876693687185840, 11.61506469206102314030121577681

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