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Properties

Label 2-38-19.16-c5-0-5
Degree 22
Conductor 3838
Sign 0.136+0.990i0.136 + 0.990i
Analytic cond. 6.094586.09458
Root an. cond. 2.468722.46872
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.06 + 2.57i)2-s + (−24.7 + 9.01i)3-s + (2.77 + 15.7i)4-s + (14.6 − 83.2i)5-s + (−99.1 − 36.0i)6-s + (−7.63 − 13.2i)7-s + (−32.0 + 55.4i)8-s + (346. − 290. i)9-s + (258. − 217. i)10-s + (225. − 390. i)11-s + (−210. − 365. i)12-s + (−754. − 274. i)13-s + (10.5 − 60.1i)14-s + (386. + 2.19e3i)15-s + (−240. + 87.5i)16-s + (−1.01e3 − 855. i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−1.58 + 0.578i)3-s + (0.0868 + 0.492i)4-s + (0.262 − 1.48i)5-s + (−1.12 − 0.409i)6-s + (−0.0588 − 0.101i)7-s + (−0.176 + 0.306i)8-s + (1.42 − 1.19i)9-s + (0.819 − 0.687i)10-s + (0.561 − 0.971i)11-s + (−0.422 − 0.732i)12-s + (−1.23 − 0.450i)13-s + (0.0144 − 0.0819i)14-s + (0.444 + 2.51i)15-s + (−0.234 + 0.0855i)16-s + (−0.855 − 0.717i)17-s + ⋯

Functional equation

Λ(s)=(38s/2 ΓC(s) L(s)=((0.136+0.990i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(38s/2 ΓC(s+5/2) L(s)=((0.136+0.990i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3838    =    2⋅192 \cdot 19
Sign: 0.136+0.990i0.136 + 0.990i
Analytic conductor: 6.094586.09458
Root analytic conductor: 2.468722.46872
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ38(35,⋅)\chi_{38} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 38, ( :5/2), 0.136+0.990i)(2,\ 38,\ (\ :5/2),\ 0.136 + 0.990i)

Particular Values

L(3)L(3) ≈\approx 0.588287−0.512851i0.588287 - 0.512851i
L(12)L(\frac12) ≈\approx 0.588287−0.512851i0.588287 - 0.512851i
L(72)L(\frac{7}{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+(−3.06−2.57i)T 1 + (-3.06 - 2.57i)T
19 1+(1.50e3−459.i)T 1 + (1.50e3 - 459. i)T
good3 1+(24.7−9.01i)T+(186.−156.i)T2 1 + (24.7 - 9.01i)T + (186. - 156. i)T^{2}
5 1+(−14.6+83.2i)T+(−2.93e3−1.06e3i)T2 1 + (-14.6 + 83.2i)T + (-2.93e3 - 1.06e3i)T^{2}
7 1+(7.63+13.2i)T+(−8.40e3+1.45e4i)T2 1 + (7.63 + 13.2i)T + (-8.40e3 + 1.45e4i)T^{2}
11 1+(−225.+390.i)T+(−8.05e4−1.39e5i)T2 1 + (-225. + 390. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+(754.+274.i)T+(2.84e5+2.38e5i)T2 1 + (754. + 274. i)T + (2.84e5 + 2.38e5i)T^{2}
17 1+(1.01e3+855.i)T+(2.46e5+1.39e6i)T2 1 + (1.01e3 + 855. i)T + (2.46e5 + 1.39e6i)T^{2}
23 1+(−65.9−373.i)T+(−6.04e6+2.20e6i)T2 1 + (-65.9 - 373. i)T + (-6.04e6 + 2.20e6i)T^{2}
29 1+(−6.38e3+5.35e3i)T+(3.56e6−2.01e7i)T2 1 + (-6.38e3 + 5.35e3i)T + (3.56e6 - 2.01e7i)T^{2}
31 1+(−3.34e3−5.79e3i)T+(−1.43e7+2.47e7i)T2 1 + (-3.34e3 - 5.79e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+4.85e3T+6.93e7T2 1 + 4.85e3T + 6.93e7T^{2}
41 1+(1.18e4−4.30e3i)T+(8.87e7−7.44e7i)T2 1 + (1.18e4 - 4.30e3i)T + (8.87e7 - 7.44e7i)T^{2}
43 1+(−2.55e3+1.45e4i)T+(−1.38e8−5.02e7i)T2 1 + (-2.55e3 + 1.45e4i)T + (-1.38e8 - 5.02e7i)T^{2}
47 1+(9.45e3−7.92e3i)T+(3.98e7−2.25e8i)T2 1 + (9.45e3 - 7.92e3i)T + (3.98e7 - 2.25e8i)T^{2}
53 1+(−286.−1.62e3i)T+(−3.92e8+1.43e8i)T2 1 + (-286. - 1.62e3i)T + (-3.92e8 + 1.43e8i)T^{2}
59 1+(−4.24e3−3.55e3i)T+(1.24e8+7.04e8i)T2 1 + (-4.24e3 - 3.55e3i)T + (1.24e8 + 7.04e8i)T^{2}
61 1+(−5.83e3−3.31e4i)T+(−7.93e8+2.88e8i)T2 1 + (-5.83e3 - 3.31e4i)T + (-7.93e8 + 2.88e8i)T^{2}
67 1+(−824.+691.i)T+(2.34e8−1.32e9i)T2 1 + (-824. + 691. i)T + (2.34e8 - 1.32e9i)T^{2}
71 1+(−4.28e3+2.42e4i)T+(−1.69e9−6.17e8i)T2 1 + (-4.28e3 + 2.42e4i)T + (-1.69e9 - 6.17e8i)T^{2}
73 1+(−4.67e4+1.70e4i)T+(1.58e9−1.33e9i)T2 1 + (-4.67e4 + 1.70e4i)T + (1.58e9 - 1.33e9i)T^{2}
79 1+(2.89e4−1.05e4i)T+(2.35e9−1.97e9i)T2 1 + (2.89e4 - 1.05e4i)T + (2.35e9 - 1.97e9i)T^{2}
83 1+(1.88e4+3.25e4i)T+(−1.96e9+3.41e9i)T2 1 + (1.88e4 + 3.25e4i)T + (-1.96e9 + 3.41e9i)T^{2}
89 1+(−8.71e4−3.17e4i)T+(4.27e9+3.58e9i)T2 1 + (-8.71e4 - 3.17e4i)T + (4.27e9 + 3.58e9i)T^{2}
97 1+(8.78e4+7.37e4i)T+(1.49e9+8.45e9i)T2 1 + (8.78e4 + 7.37e4i)T + (1.49e9 + 8.45e9i)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

−15.45722533613168473495995629542, −13.65976167669869281400378221855, −12.40804618436842582355030304218, −11.72784761793478615161570717778, −10.17150156461963839842113773090, −8.672560494218737899361745267430, −6.51648878261310160418284438603, −5.28475417705690817684029230914, −4.48476188896600499318283939100, −0.44145639808686111498609112641, 2.13646882255354498416696226385, 4.67826582885240886180305684776, 6.44240924159415283368875183907, 6.89976819340604751811454783918, 10.04286037450898769593208720836, 10.90944112396902150549610708413, 11.90535475653944478446556014140, 12.80340514078238823927251138414, 14.33448749049354330896140947097, 15.31759380163255995715695640221

Graph of the ZZ-function along the critical line