Properties

Label 2-380-95.13-c1-0-7
Degree $2$
Conductor $380$
Sign $-0.634 + 0.772i$
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  + (−0.927 + 0.432i)3-s + (0.140 − 2.23i)5-s + (−0.231 − 0.864i)7-s + (−1.25 + 1.49i)9-s + (−3.29 − 5.70i)11-s + (−2.04 + 4.38i)13-s + (0.834 + 2.13i)15-s + (0.154 − 1.76i)17-s + (−3.61 − 2.43i)19-s + (0.588 + 0.701i)21-s + (5.10 − 3.57i)23-s + (−4.96 − 0.628i)25-s + (1.31 − 4.89i)27-s + (−3.21 − 2.70i)29-s + (−2.00 − 1.15i)31-s + ⋯
L(s)  = 1  + (−0.535 + 0.249i)3-s + (0.0630 − 0.998i)5-s + (−0.0875 − 0.326i)7-s + (−0.418 + 0.498i)9-s + (−0.992 − 1.71i)11-s + (−0.567 + 1.21i)13-s + (0.215 + 0.549i)15-s + (0.0373 − 0.427i)17-s + (−0.828 − 0.559i)19-s + (0.128 + 0.153i)21-s + (1.06 − 0.746i)23-s + (−0.992 − 0.125i)25-s + (0.252 − 0.942i)27-s + (−0.597 − 0.501i)29-s + (−0.360 − 0.207i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.634 + 0.772i$
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: \((2,\ 380,\ (\ :1/2),\ -0.634 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252450 - 0.534139i\)
\(L(\frac12)\) \(\approx\) \(0.252450 - 0.534139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.140 + 2.23i)T \)
19 \( 1 + (3.61 + 2.43i)T \)
good3 \( 1 + (0.927 - 0.432i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (0.231 + 0.864i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.29 + 5.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.04 - 4.38i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (-0.154 + 1.76i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (-5.10 + 3.57i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (3.21 + 2.70i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.00 + 1.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.30 + 3.30i)T + 37iT^{2} \)
41 \( 1 + (2.25 - 6.19i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.82 + 2.60i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (-5.57 + 0.488i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (-6.39 - 9.12i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (-1.36 + 1.14i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.13 + 6.45i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.953 - 10.8i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (10.2 - 1.81i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-3.06 - 6.56i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (1.84 + 0.672i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (15.1 - 4.06i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-13.1 + 4.79i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-12.2 - 1.07i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.10765250243709046177815806359, −10.28536095057574299939224506319, −8.996426023092321556339529530755, −8.488179409474060110805672780337, −7.27192485045885216759070926681, −5.93934019889706361401855896860, −5.16805726869069354299760529482, −4.24257791399163486938836121023, −2.53105792901843482570910265367, −0.39475812676732117232189222553, 2.21913918091248905927916569051, 3.40312117867846642318889475540, 5.09100239498455490851117076247, 5.90164925621163325212845565496, 7.06145241833778304703236000568, 7.62413693656673206028711980504, 9.008049553471100238516756062534, 10.23656825280434676794303388827, 10.55421277114691785918239538263, 11.74028278186961488959001607439

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