Properties

Label 2-380-380.219-c1-0-44
Degree $2$
Conductor $380$
Sign $0.0488 + 0.998i$
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  + (−1.31 − 0.531i)2-s + (0.664 + 0.791i)3-s + (1.43 + 1.39i)4-s + (0.846 − 2.06i)5-s + (−0.450 − 1.39i)6-s + (1.98 − 3.43i)7-s + (−1.14 − 2.58i)8-s + (0.335 − 1.90i)9-s + (−2.20 + 2.26i)10-s + (−3.95 + 2.28i)11-s + (−0.149 + 2.06i)12-s + (−1.59 − 1.34i)13-s + (−4.42 + 3.44i)14-s + (2.20 − 0.704i)15-s + (0.120 + 3.99i)16-s + (−3.02 + 0.532i)17-s + ⋯
L(s)  = 1  + (−0.926 − 0.375i)2-s + (0.383 + 0.457i)3-s + (0.717 + 0.696i)4-s + (0.378 − 0.925i)5-s + (−0.183 − 0.567i)6-s + (0.749 − 1.29i)7-s + (−0.403 − 0.915i)8-s + (0.111 − 0.634i)9-s + (−0.698 + 0.715i)10-s + (−1.19 + 0.689i)11-s + (−0.0430 + 0.595i)12-s + (−0.442 − 0.371i)13-s + (−1.18 + 0.921i)14-s + (0.568 − 0.182i)15-s + (0.0300 + 0.999i)16-s + (−0.732 + 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0488 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0488 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0488 + 0.998i$
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: \((2,\ 380,\ (\ :1/2),\ 0.0488 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738487 - 0.703257i\)
\(L(\frac12)\) \(\approx\) \(0.738487 - 0.703257i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.531i)T \)
5 \( 1 + (-0.846 + 2.06i)T \)
19 \( 1 + (-2.31 + 3.69i)T \)
good3 \( 1 + (-0.664 - 0.791i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (-1.98 + 3.43i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.95 - 2.28i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.59 + 1.34i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (3.02 - 0.532i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (4.89 - 1.78i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-8.07 - 1.42i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-0.128 + 0.223i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.87T + 37T^{2} \)
41 \( 1 + (-3.49 - 4.16i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.50 + 1.27i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.32 - 7.54i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-5.94 + 2.16i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.0505 - 0.286i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-12.9 + 4.72i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-6.61 - 1.16i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-4.32 - 1.57i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.91 - 5.85i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-1.37 + 1.15i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.10 + 5.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.410 + 0.489i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-3.18 - 18.0i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.82837483020900148080701307332, −10.00591114700846120127379623430, −9.548422509099917402536675351071, −8.363041072277967109948974277458, −7.77828307600962297235325350811, −6.69822189764646896892849235608, −4.94780581061117866232601654232, −4.06922674573588023352950816860, −2.47798266082917076085350899764, −0.882371924769916065371463775211, 2.12716131801367150974809516498, 2.59473485709930834146725866197, 5.16322507279029755243153481288, 5.97777585818470848949240199790, 7.07321088259954921900316177513, 8.110850065437173395449961624129, 8.433286573275931460393891252201, 9.750502743587796091421499486511, 10.53683814707334878845460515898, 11.35252356294817540997582183865

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