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Properties

Label 2-38-19.5-c5-0-8
Degree 22
Conductor 3838
Sign −0.990−0.134i-0.990 - 0.134i
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  + (−3.75 + 1.36i)2-s + (4.41 − 25.0i)3-s + (12.2 − 10.2i)4-s + (−71.5 − 60.0i)5-s + (17.6 + 100. i)6-s + (91.0 + 157. i)7-s + (−32.0 + 55.4i)8-s + (−377. − 137. i)9-s + (351. + 127. i)10-s + (−198. + 344. i)11-s + (−203. − 351. i)12-s + (−119. − 680. i)13-s + (−558. − 468. i)14-s + (−1.81e3 + 1.52e3i)15-s + (44.4 − 252. i)16-s + (−1.15e3 + 421. i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.282 − 1.60i)3-s + (0.383 − 0.321i)4-s + (−1.28 − 1.07i)5-s + (0.200 + 1.13i)6-s + (0.702 + 1.21i)7-s + (−0.176 + 0.306i)8-s + (−1.55 − 0.565i)9-s + (1.11 + 0.404i)10-s + (−0.495 + 0.858i)11-s + (−0.407 − 0.705i)12-s + (−0.196 − 1.11i)13-s + (−0.761 − 0.638i)14-s + (−2.08 + 1.75i)15-s + (0.0434 − 0.246i)16-s + (−0.972 + 0.353i)17-s + ⋯

Functional equation

Λ(s)=(38s/2 ΓC(s) L(s)=((−0.990−0.134i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(38s/2 ΓC(s+5/2) L(s)=((−0.990−0.134i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3838    =    2⋅192 \cdot 19
Sign: −0.990−0.134i-0.990 - 0.134i
Analytic conductor: 6.094586.09458
Root analytic conductor: 2.468722.46872
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ38(5,⋅)\chi_{38} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 38, ( :5/2), −0.990−0.134i)(2,\ 38,\ (\ :5/2),\ -0.990 - 0.134i)

Particular Values

L(3)L(3) ≈\approx 0.0369392+0.547817i0.0369392 + 0.547817i
L(12)L(\frac12) ≈\approx 0.0369392+0.547817i0.0369392 + 0.547817i
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+(3.75−1.36i)T 1 + (3.75 - 1.36i)T
19 1+(−785.+1.36e3i)T 1 + (-785. + 1.36e3i)T
good3 1+(−4.41+25.0i)T+(−228.−83.1i)T2 1 + (-4.41 + 25.0i)T + (-228. - 83.1i)T^{2}
5 1+(71.5+60.0i)T+(542.+3.07e3i)T2 1 + (71.5 + 60.0i)T + (542. + 3.07e3i)T^{2}
7 1+(−91.0−157.i)T+(−8.40e3+1.45e4i)T2 1 + (-91.0 - 157. i)T + (-8.40e3 + 1.45e4i)T^{2}
11 1+(198.−344.i)T+(−8.05e4−1.39e5i)T2 1 + (198. - 344. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+(119.+680.i)T+(−3.48e5+1.26e5i)T2 1 + (119. + 680. i)T + (-3.48e5 + 1.26e5i)T^{2}
17 1+(1.15e3−421.i)T+(1.08e6−9.12e5i)T2 1 + (1.15e3 - 421. i)T + (1.08e6 - 9.12e5i)T^{2}
23 1+(−216.+181.i)T+(1.11e6−6.33e6i)T2 1 + (-216. + 181. i)T + (1.11e6 - 6.33e6i)T^{2}
29 1+(4.07e3+1.48e3i)T+(1.57e7+1.31e7i)T2 1 + (4.07e3 + 1.48e3i)T + (1.57e7 + 1.31e7i)T^{2}
31 1+(1.34e3+2.32e3i)T+(−1.43e7+2.47e7i)T2 1 + (1.34e3 + 2.32e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+3.85e3T+6.93e7T2 1 + 3.85e3T + 6.93e7T^{2}
41 1+(−3.39e3+1.92e4i)T+(−1.08e8−3.96e7i)T2 1 + (-3.39e3 + 1.92e4i)T + (-1.08e8 - 3.96e7i)T^{2}
43 1+(284.+238.i)T+(2.55e7+1.44e8i)T2 1 + (284. + 238. i)T + (2.55e7 + 1.44e8i)T^{2}
47 1+(1.27e4+4.62e3i)T+(1.75e8+1.47e8i)T2 1 + (1.27e4 + 4.62e3i)T + (1.75e8 + 1.47e8i)T^{2}
53 1+(−4.30e3+3.61e3i)T+(7.26e7−4.11e8i)T2 1 + (-4.30e3 + 3.61e3i)T + (7.26e7 - 4.11e8i)T^{2}
59 1+(−2.82e4+1.02e4i)T+(5.47e8−4.59e8i)T2 1 + (-2.82e4 + 1.02e4i)T + (5.47e8 - 4.59e8i)T^{2}
61 1+(−1.69e3+1.42e3i)T+(1.46e8−8.31e8i)T2 1 + (-1.69e3 + 1.42e3i)T + (1.46e8 - 8.31e8i)T^{2}
67 1+(249.+90.9i)T+(1.03e9+8.67e8i)T2 1 + (249. + 90.9i)T + (1.03e9 + 8.67e8i)T^{2}
71 1+(−1.70e4−1.43e4i)T+(3.13e8+1.77e9i)T2 1 + (-1.70e4 - 1.43e4i)T + (3.13e8 + 1.77e9i)T^{2}
73 1+(−9.67e3+5.48e4i)T+(−1.94e9−7.09e8i)T2 1 + (-9.67e3 + 5.48e4i)T + (-1.94e9 - 7.09e8i)T^{2}
79 1+(9.82e3−5.56e4i)T+(−2.89e9−1.05e9i)T2 1 + (9.82e3 - 5.56e4i)T + (-2.89e9 - 1.05e9i)T^{2}
83 1+(4.25e4+7.37e4i)T+(−1.96e9+3.41e9i)T2 1 + (4.25e4 + 7.37e4i)T + (-1.96e9 + 3.41e9i)T^{2}
89 1+(−3.76e3−2.13e4i)T+(−5.24e9+1.90e9i)T2 1 + (-3.76e3 - 2.13e4i)T + (-5.24e9 + 1.90e9i)T^{2}
97 1+(1.04e5−3.81e4i)T+(6.57e9−5.51e9i)T2 1 + (1.04e5 - 3.81e4i)T + (6.57e9 - 5.51e9i)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

−15.04700416368871650008378681809, −13.05375394406036130039669388319, −12.32355872036585432440478394819, −11.40306466011049597847768896106, −8.888512033334952208172976263938, −8.116183385245456185955599826303, −7.29301102253709635053351313188, −5.28770559154756537252041486266, −2.11731036099219031010878801531, −0.36574411998479145561017828968, 3.34441290498922989242804398315, 4.35963027130056289785619609867, 7.21729484063745164372987231783, 8.412801509284820722389892013391, 9.937256831823726118764289827146, 11.10762207484461834910326450345, 11.25985204965013986325514132398, 14.04895915743965120285991728817, 14.91379321931548100488633390127, 16.05413822901568666482661009044

Graph of the ZZ-function along the critical line