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Properties

Label 2-380-19.7-c1-0-5
Degree 22
Conductor 380380
Sign 0.360+0.932i0.360 + 0.932i
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  + (−0.354 − 0.614i)3-s + (−0.5 − 0.866i)5-s + 3.11·7-s + (1.24 − 2.16i)9-s − 3.52·11-s + (−0.200 + 0.347i)13-s + (−0.354 + 0.614i)15-s + (−1.74 − 3.02i)17-s + (4.35 − 0.251i)19-s + (−1.10 − 1.91i)21-s + (3.65 − 6.33i)23-s + (−0.499 + 0.866i)25-s − 3.89·27-s + (3.96 − 6.86i)29-s + 5.73·31-s + ⋯
L(s)  = 1  + (−0.204 − 0.354i)3-s + (−0.223 − 0.387i)5-s + 1.17·7-s + (0.416 − 0.720i)9-s − 1.06·11-s + (−0.0556 + 0.0964i)13-s + (−0.0915 + 0.158i)15-s + (−0.424 − 0.734i)17-s + (0.998 − 0.0577i)19-s + (−0.240 − 0.416i)21-s + (0.762 − 1.32i)23-s + (−0.0999 + 0.173i)25-s − 0.750·27-s + (0.736 − 1.27i)29-s + 1.03·31-s + ⋯

Functional equation

Λ(s)=(380s/2 ΓC(s) L(s)=((0.360+0.932i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(380s/2 ΓC(s+1/2) L(s)=((0.360+0.932i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 380380    =    22⋅5⋅192^{2} \cdot 5 \cdot 19
Sign: 0.360+0.932i0.360 + 0.932i
Analytic conductor: 3.034313.03431
Root analytic conductor: 1.741921.74192
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ380(121,⋅)\chi_{380} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 380, ( :1/2), 0.360+0.932i)(2,\ 380,\ (\ :1/2),\ 0.360 + 0.932i)

Particular Values

L(1)L(1) ≈\approx 1.07532−0.737442i1.07532 - 0.737442i
L(12)L(\frac12) ≈\approx 1.07532−0.737442i1.07532 - 0.737442i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(−4.35+0.251i)T 1 + (-4.35 + 0.251i)T
good3 1+(0.354+0.614i)T+(−1.5+2.59i)T2 1 + (0.354 + 0.614i)T + (-1.5 + 2.59i)T^{2}
7 1−3.11T+7T2 1 - 3.11T + 7T^{2}
11 1+3.52T+11T2 1 + 3.52T + 11T^{2}
13 1+(0.200−0.347i)T+(−6.5−11.2i)T2 1 + (0.200 - 0.347i)T + (-6.5 - 11.2i)T^{2}
17 1+(1.74+3.02i)T+(−8.5+14.7i)T2 1 + (1.74 + 3.02i)T + (-8.5 + 14.7i)T^{2}
23 1+(−3.65+6.33i)T+(−11.5−19.9i)T2 1 + (-3.65 + 6.33i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.96+6.86i)T+(−14.5−25.1i)T2 1 + (-3.96 + 6.86i)T + (-14.5 - 25.1i)T^{2}
31 1−5.73T+31T2 1 - 5.73T + 31T^{2}
37 1+10.5T+37T2 1 + 10.5T + 37T^{2}
41 1+(−0.555−0.961i)T+(−20.5+35.5i)T2 1 + (-0.555 - 0.961i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4.30−7.45i)T+(−21.5+37.2i)T2 1 + (-4.30 - 7.45i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.76−6.51i)T+(−23.5−40.7i)T2 1 + (3.76 - 6.51i)T + (-23.5 - 40.7i)T^{2}
53 1+(5.27−9.14i)T+(−26.5−45.8i)T2 1 + (5.27 - 9.14i)T + (-26.5 - 45.8i)T^{2}
59 1+(−4.25−7.37i)T+(−29.5+51.0i)T2 1 + (-4.25 - 7.37i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.61+7.98i)T+(−30.5−52.8i)T2 1 + (-4.61 + 7.98i)T + (-30.5 - 52.8i)T^{2}
67 1+(4.20−7.28i)T+(−33.5−58.0i)T2 1 + (4.20 - 7.28i)T + (-33.5 - 58.0i)T^{2}
71 1+(−4.31−7.46i)T+(−35.5+61.4i)T2 1 + (-4.31 - 7.46i)T + (-35.5 + 61.4i)T^{2}
73 1+(0.870+1.50i)T+(−36.5+63.2i)T2 1 + (0.870 + 1.50i)T + (-36.5 + 63.2i)T^{2}
79 1+(−4.50−7.80i)T+(−39.5+68.4i)T2 1 + (-4.50 - 7.80i)T + (-39.5 + 68.4i)T^{2}
83 1−4.07T+83T2 1 - 4.07T + 83T^{2}
89 1+(−6.61+11.4i)T+(−44.5−77.0i)T2 1 + (-6.61 + 11.4i)T + (-44.5 - 77.0i)T^{2}
97 1+(−3.85−6.67i)T+(−48.5+84.0i)T2 1 + (-3.85 - 6.67i)T + (-48.5 + 84.0i)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

−11.36130138634138808513985031500, −10.35339171649876155706031931121, −9.291921410225189693146220660729, −8.250419959199405651928533687454, −7.53800808556912802475421025199, −6.47727010442478896766648228469, −5.13996140523067507593566820827, −4.44402740886320771401966039019, −2.68729767403263981844063335742, −0.990413314656452807125653544747, 1.84213141740888498894496777583, 3.41937975074693589557824707093, 4.89827486425458345349048421789, 5.32531640845324913121052485521, 7.01216042936862670405579627020, 7.80636194875614772006068489193, 8.594068927210827818089321642360, 10.00015561887179724818012301001, 10.69957960919140263787281294738, 11.28028638652935778823611564418

Graph of the ZZ-function along the critical line