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Properties

Label 2-380-95.13-c1-0-2
Degree $2$
Conductor $380$
Sign $0.655 - 0.754i$
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  + (−2.74 + 1.28i)3-s + (−1.16 − 1.91i)5-s + (−0.370 − 1.38i)7-s + (3.97 − 4.74i)9-s + (2.73 + 4.73i)11-s + (−0.160 + 0.345i)13-s + (5.63 + 3.76i)15-s + (−0.0144 + 0.165i)17-s + (1.45 + 4.10i)19-s + (2.78 + 3.32i)21-s + (0.363 − 0.254i)23-s + (−2.30 + 4.43i)25-s + (−2.50 + 9.34i)27-s + (1.09 + 0.921i)29-s + (6.67 + 3.85i)31-s + ⋯
L(s)  = 1  + (−1.58 + 0.739i)3-s + (−0.518 − 0.854i)5-s + (−0.139 − 0.522i)7-s + (1.32 − 1.58i)9-s + (0.823 + 1.42i)11-s + (−0.0446 + 0.0956i)13-s + (1.45 + 0.972i)15-s + (−0.00350 + 0.0400i)17-s + (0.333 + 0.942i)19-s + (0.608 + 0.725i)21-s + (0.0757 − 0.0530i)23-s + (−0.461 + 0.887i)25-s + (−0.481 + 1.79i)27-s + (0.203 + 0.171i)29-s + (1.19 + 0.692i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.655 - 0.754i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.655 - 0.754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633660 + 0.288822i\)
\(L(\frac12)\) \(\approx\) \(0.633660 + 0.288822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 + 1.91i)T \)
19 \( 1 + (-1.45 - 4.10i)T \)
good3 \( 1 + (2.74 - 1.28i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (0.370 + 1.38i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.73 - 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.160 - 0.345i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (0.0144 - 0.165i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (-0.363 + 0.254i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (-1.09 - 0.921i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-6.67 - 3.85i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.77 - 6.77i)T + 37iT^{2} \)
41 \( 1 + (-2.40 + 6.59i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-6.61 + 9.44i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (3.11 - 0.272i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (-6.56 - 9.37i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (8.39 - 7.04i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.07 + 11.7i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.773 - 8.83i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (0.325 - 0.0574i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (0.964 + 2.06i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (2.57 + 0.936i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (11.1 - 2.97i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (5.02 - 1.82i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.497 - 0.0434i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.69562786219585725897010529410, −10.49479117035280482937634349202, −9.918086639491587145831439608593, −8.992271380754875588168872667572, −7.52348653622149563472024970472, −6.60566337420982112216496287030, −5.49873119427901626507066656560, −4.50476129439488948748900228992, −4.00800581263571277891264695129, −1.11500515402580346062516976736, 0.74154449840458013873839406800, 2.81546732557625040652565138585, 4.40913949006113693617639772167, 5.84912087092265138098957289648, 6.27113686826970046613992174076, 7.19687697557512498999535405490, 8.199742164501140114049227477127, 9.560484383717466354072830682440, 10.82326352778189654965991289237, 11.40750696096432969912337216281

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