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Properties

Label 2-380-19.4-c1-0-4
Degree 22
Conductor 380380
Sign −0.129+0.991i-0.129 + 0.991i
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  + (−0.275 − 1.56i)3-s + (0.766 − 0.642i)5-s + (0.778 − 1.34i)7-s + (0.454 − 0.165i)9-s + (−1.44 − 2.50i)11-s + (−0.501 + 2.84i)13-s + (−1.21 − 1.01i)15-s + (−3.43 − 1.25i)17-s + (3.58 − 2.48i)19-s + (−2.32 − 0.845i)21-s + (−1.02 − 0.860i)23-s + (0.173 − 0.984i)25-s + (−2.76 − 4.78i)27-s + (4.25 − 1.54i)29-s + (−0.0994 + 0.172i)31-s + ⋯
L(s)  = 1  + (−0.159 − 0.902i)3-s + (0.342 − 0.287i)5-s + (0.294 − 0.509i)7-s + (0.151 − 0.0550i)9-s + (−0.435 − 0.754i)11-s + (−0.139 + 0.788i)13-s + (−0.313 − 0.263i)15-s + (−0.834 − 0.303i)17-s + (0.821 − 0.570i)19-s + (−0.506 − 0.184i)21-s + (−0.213 − 0.179i)23-s + (0.0347 − 0.196i)25-s + (−0.531 − 0.920i)27-s + (0.789 − 0.287i)29-s + (−0.0178 + 0.0309i)31-s + ⋯

Functional equation

Λ(s)=(380s/2 ΓC(s) L(s)=((−0.129+0.991i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(380s/2 ΓC(s+1/2) L(s)=((−0.129+0.991i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 380380    =    22⋅5⋅192^{2} \cdot 5 \cdot 19
Sign: −0.129+0.991i-0.129 + 0.991i
Analytic conductor: 3.034313.03431
Root analytic conductor: 1.741921.74192
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ380(61,⋅)\chi_{380} (61, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 380, ( :1/2), −0.129+0.991i)(2,\ 380,\ (\ :1/2),\ -0.129 + 0.991i)

Particular Values

L(1)L(1) ≈\approx 0.881518−1.00453i0.881518 - 1.00453i
L(12)L(\frac12) ≈\approx 0.881518−1.00453i0.881518 - 1.00453i
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
5 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
19 1+(−3.58+2.48i)T 1 + (-3.58 + 2.48i)T
good3 1+(0.275+1.56i)T+(−2.81+1.02i)T2 1 + (0.275 + 1.56i)T + (-2.81 + 1.02i)T^{2}
7 1+(−0.778+1.34i)T+(−3.5−6.06i)T2 1 + (-0.778 + 1.34i)T + (-3.5 - 6.06i)T^{2}
11 1+(1.44+2.50i)T+(−5.5+9.52i)T2 1 + (1.44 + 2.50i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.501−2.84i)T+(−12.2−4.44i)T2 1 + (0.501 - 2.84i)T + (-12.2 - 4.44i)T^{2}
17 1+(3.43+1.25i)T+(13.0+10.9i)T2 1 + (3.43 + 1.25i)T + (13.0 + 10.9i)T^{2}
23 1+(1.02+0.860i)T+(3.99+22.6i)T2 1 + (1.02 + 0.860i)T + (3.99 + 22.6i)T^{2}
29 1+(−4.25+1.54i)T+(22.2−18.6i)T2 1 + (-4.25 + 1.54i)T + (22.2 - 18.6i)T^{2}
31 1+(0.0994−0.172i)T+(−15.5−26.8i)T2 1 + (0.0994 - 0.172i)T + (-15.5 - 26.8i)T^{2}
37 1+6.14T+37T2 1 + 6.14T + 37T^{2}
41 1+(−0.240−1.36i)T+(−38.5+14.0i)T2 1 + (-0.240 - 1.36i)T + (-38.5 + 14.0i)T^{2}
43 1+(−1.02+0.861i)T+(7.46−42.3i)T2 1 + (-1.02 + 0.861i)T + (7.46 - 42.3i)T^{2}
47 1+(1.00−0.366i)T+(36.0−30.2i)T2 1 + (1.00 - 0.366i)T + (36.0 - 30.2i)T^{2}
53 1+(−5.96−5.00i)T+(9.20+52.1i)T2 1 + (-5.96 - 5.00i)T + (9.20 + 52.1i)T^{2}
59 1+(−4.57−1.66i)T+(45.1+37.9i)T2 1 + (-4.57 - 1.66i)T + (45.1 + 37.9i)T^{2}
61 1+(−7.26−6.09i)T+(10.5+60.0i)T2 1 + (-7.26 - 6.09i)T + (10.5 + 60.0i)T^{2}
67 1+(−7.36+2.68i)T+(51.3−43.0i)T2 1 + (-7.36 + 2.68i)T + (51.3 - 43.0i)T^{2}
71 1+(−1.21+1.02i)T+(12.3−69.9i)T2 1 + (-1.21 + 1.02i)T + (12.3 - 69.9i)T^{2}
73 1+(−2.01−11.4i)T+(−68.5+24.9i)T2 1 + (-2.01 - 11.4i)T + (-68.5 + 24.9i)T^{2}
79 1+(−2.38−13.5i)T+(−74.2+27.0i)T2 1 + (-2.38 - 13.5i)T + (-74.2 + 27.0i)T^{2}
83 1+(−5.84+10.1i)T+(−41.5−71.8i)T2 1 + (-5.84 + 10.1i)T + (-41.5 - 71.8i)T^{2}
89 1+(2.54−14.4i)T+(−83.6−30.4i)T2 1 + (2.54 - 14.4i)T + (-83.6 - 30.4i)T^{2}
97 1+(0.990+0.360i)T+(74.3+62.3i)T2 1 + (0.990 + 0.360i)T + (74.3 + 62.3i)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

−11.27223878268465338033190225327, −10.24500181812451582149622605705, −9.200319219498462402936378653157, −8.237158493992482828423537922633, −7.20034983584266748588889962068, −6.53517948109928316944572769069, −5.31889912830912896785960148951, −4.15389739339759859109569725829, −2.41650122820220261354097770447, −0.973920049023747323381824895310, 2.10338935061628216512316786486, 3.56683132656891231639246382393, 4.85922325379728790549941884514, 5.51894443152191514538584294636, 6.86367009834831307519157020446, 7.937063924529617612693000975656, 9.021935525783366853009398075949, 10.05510355106884000045819882252, 10.39493294690944895461755582469, 11.45586570745066665466083604605

Graph of the ZZ-function along the critical line