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Properties

Label 2-380-380.59-c1-0-12
Degree 22
Conductor 380380
Sign 0.0488−0.998i0.0488 - 0.998i
Analytic cond. 3.034313.03431
Root an. cond. 1.741921.74192
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(380s/2 ΓC(s) L(s)=((0.0488−0.998i) Λ‾(2−s)\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}
Λ(s)=(380s/2 ΓC(s+1/2) L(s)=((0.0488−0.998i) Λ‾(1−s)\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: 22
Conductor: 380380    =    22⋅5⋅192^{2} \cdot 5 \cdot 19
Sign: 0.0488−0.998i0.0488 - 0.998i
Analytic conductor: 3.034313.03431
Root analytic conductor: 1.741921.74192
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ380(59,⋅)\chi_{380} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 380, ( :1/2), 0.0488−0.998i)(2,\ 380,\ (\ :1/2),\ 0.0488 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 0.738487+0.703257i0.738487 + 0.703257i
L(12)L(\frac12) ≈\approx 0.738487+0.703257i0.738487 + 0.703257i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.31−0.531i)T 1 + (1.31 - 0.531i)T
5 1+(−0.846−2.06i)T 1 + (-0.846 - 2.06i)T
19 1+(−2.31−3.69i)T 1 + (-2.31 - 3.69i)T
good3 1+(−0.664+0.791i)T+(−0.520−2.95i)T2 1 + (-0.664 + 0.791i)T + (-0.520 - 2.95i)T^{2}
7 1+(−1.98−3.43i)T+(−3.5+6.06i)T2 1 + (-1.98 - 3.43i)T + (-3.5 + 6.06i)T^{2}
11 1+(3.95+2.28i)T+(5.5+9.52i)T2 1 + (3.95 + 2.28i)T + (5.5 + 9.52i)T^{2}
13 1+(1.59−1.34i)T+(2.25−12.8i)T2 1 + (1.59 - 1.34i)T + (2.25 - 12.8i)T^{2}
17 1+(3.02+0.532i)T+(15.9+5.81i)T2 1 + (3.02 + 0.532i)T + (15.9 + 5.81i)T^{2}
23 1+(4.89+1.78i)T+(17.6+14.7i)T2 1 + (4.89 + 1.78i)T + (17.6 + 14.7i)T^{2}
29 1+(−8.07+1.42i)T+(27.2−9.91i)T2 1 + (-8.07 + 1.42i)T + (27.2 - 9.91i)T^{2}
31 1+(−0.128−0.223i)T+(−15.5+26.8i)T2 1 + (-0.128 - 0.223i)T + (-15.5 + 26.8i)T^{2}
37 1−4.87T+37T2 1 - 4.87T + 37T^{2}
41 1+(−3.49+4.16i)T+(−7.11−40.3i)T2 1 + (-3.49 + 4.16i)T + (-7.11 - 40.3i)T^{2}
43 1+(3.50−1.27i)T+(32.9−27.6i)T2 1 + (3.50 - 1.27i)T + (32.9 - 27.6i)T^{2}
47 1+(1.32+7.54i)T+(−44.1+16.0i)T2 1 + (1.32 + 7.54i)T + (-44.1 + 16.0i)T^{2}
53 1+(−5.94−2.16i)T+(40.6+34.0i)T2 1 + (-5.94 - 2.16i)T + (40.6 + 34.0i)T^{2}
59 1+(−0.0505+0.286i)T+(−55.4−20.1i)T2 1 + (-0.0505 + 0.286i)T + (-55.4 - 20.1i)T^{2}
61 1+(−12.9−4.72i)T+(46.7+39.2i)T2 1 + (-12.9 - 4.72i)T + (46.7 + 39.2i)T^{2}
67 1+(−6.61+1.16i)T+(62.9−22.9i)T2 1 + (-6.61 + 1.16i)T + (62.9 - 22.9i)T^{2}
71 1+(−4.32+1.57i)T+(54.3−45.6i)T2 1 + (-4.32 + 1.57i)T + (54.3 - 45.6i)T^{2}
73 1+(−4.91+5.85i)T+(−12.6−71.8i)T2 1 + (-4.91 + 5.85i)T + (-12.6 - 71.8i)T^{2}
79 1+(−1.37−1.15i)T+(13.7+77.7i)T2 1 + (-1.37 - 1.15i)T + (13.7 + 77.7i)T^{2}
83 1+(−3.10−5.37i)T+(−41.5+71.8i)T2 1 + (-3.10 - 5.37i)T + (-41.5 + 71.8i)T^{2}
89 1+(−0.410−0.489i)T+(−15.4+87.6i)T2 1 + (-0.410 - 0.489i)T + (-15.4 + 87.6i)T^{2}
97 1+(−3.18+18.0i)T+(−91.1−33.1i)T2 1 + (-3.18 + 18.0i)T + (-91.1 - 33.1i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

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

Graph of the ZZ-function along the critical line