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Properties

Label 2-38-19.17-c5-0-0
Degree 22
Conductor 3838
Sign −0.0930−0.995i-0.0930 - 0.995i
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  + (0.694 − 3.93i)2-s + (−8.20 − 6.88i)3-s + (−15.0 − 5.47i)4-s + (−55.8 + 20.3i)5-s + (−32.8 + 27.5i)6-s + (115. + 199. i)7-s + (−32 + 55.4i)8-s + (−22.2 − 126. i)9-s + (41.2 + 234. i)10-s + (28.7 − 49.7i)11-s + (85.6 + 148. i)12-s + (−825. + 692. i)13-s + (866. − 315. i)14-s + (597. + 217. i)15-s + (196. + 164. i)16-s + (−149. + 850. i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.526 − 0.441i)3-s + (−0.469 − 0.171i)4-s + (−0.998 + 0.363i)5-s + (−0.372 + 0.312i)6-s + (0.888 + 1.53i)7-s + (−0.176 + 0.306i)8-s + (−0.0916 − 0.519i)9-s + (0.130 + 0.740i)10-s + (0.0716 − 0.124i)11-s + (0.171 + 0.297i)12-s + (−1.35 + 1.13i)13-s + (1.18 − 0.429i)14-s + (0.686 + 0.249i)15-s + (0.191 + 0.160i)16-s + (−0.125 + 0.713i)17-s + ⋯

Functional equation

Λ(s)=(38s/2 ΓC(s) L(s)=((−0.0930−0.995i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0930 - 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(38s/2 ΓC(s+5/2) L(s)=((−0.0930−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0930 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3838    =    2⋅192 \cdot 19
Sign: −0.0930−0.995i-0.0930 - 0.995i
Analytic conductor: 6.094586.09458
Root analytic conductor: 2.468722.46872
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ38(17,⋅)\chi_{38} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 38, ( :5/2), −0.0930−0.995i)(2,\ 38,\ (\ :5/2),\ -0.0930 - 0.995i)

Particular Values

L(3)L(3) ≈\approx 0.266575+0.292659i0.266575 + 0.292659i
L(12)L(\frac12) ≈\approx 0.266575+0.292659i0.266575 + 0.292659i
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+(−0.694+3.93i)T 1 + (-0.694 + 3.93i)T
19 1+(1.44e3+633.i)T 1 + (1.44e3 + 633. i)T
good3 1+(8.20+6.88i)T+(42.1+239.i)T2 1 + (8.20 + 6.88i)T + (42.1 + 239. i)T^{2}
5 1+(55.8−20.3i)T+(2.39e3−2.00e3i)T2 1 + (55.8 - 20.3i)T + (2.39e3 - 2.00e3i)T^{2}
7 1+(−115.−199.i)T+(−8.40e3+1.45e4i)T2 1 + (-115. - 199. i)T + (-8.40e3 + 1.45e4i)T^{2}
11 1+(−28.7+49.7i)T+(−8.05e4−1.39e5i)T2 1 + (-28.7 + 49.7i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+(825.−692.i)T+(6.44e4−3.65e5i)T2 1 + (825. - 692. i)T + (6.44e4 - 3.65e5i)T^{2}
17 1+(149.−850.i)T+(−1.33e6−4.85e5i)T2 1 + (149. - 850. i)T + (-1.33e6 - 4.85e5i)T^{2}
23 1+(1.49e3+544.i)T+(4.93e6+4.13e6i)T2 1 + (1.49e3 + 544. i)T + (4.93e6 + 4.13e6i)T^{2}
29 1+(1.25e3+7.12e3i)T+(−1.92e7+7.01e6i)T2 1 + (1.25e3 + 7.12e3i)T + (-1.92e7 + 7.01e6i)T^{2}
31 1+(601.+1.04e3i)T+(−1.43e7+2.47e7i)T2 1 + (601. + 1.04e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1−1.44e4T+6.93e7T2 1 - 1.44e4T + 6.93e7T^{2}
41 1+(−1.12e4−9.42e3i)T+(2.01e7+1.14e8i)T2 1 + (-1.12e4 - 9.42e3i)T + (2.01e7 + 1.14e8i)T^{2}
43 1+(1.63e4−5.93e3i)T+(1.12e8−9.44e7i)T2 1 + (1.63e4 - 5.93e3i)T + (1.12e8 - 9.44e7i)T^{2}
47 1+(785.+4.45e3i)T+(−2.15e8+7.84e7i)T2 1 + (785. + 4.45e3i)T + (-2.15e8 + 7.84e7i)T^{2}
53 1+(573.+208.i)T+(3.20e8+2.68e8i)T2 1 + (573. + 208. i)T + (3.20e8 + 2.68e8i)T^{2}
59 1+(1.88e3−1.07e4i)T+(−6.71e8−2.44e8i)T2 1 + (1.88e3 - 1.07e4i)T + (-6.71e8 - 2.44e8i)T^{2}
61 1+(1.59e4+5.81e3i)T+(6.46e8+5.42e8i)T2 1 + (1.59e4 + 5.81e3i)T + (6.46e8 + 5.42e8i)T^{2}
67 1+(1.30e3+7.38e3i)T+(−1.26e9+4.61e8i)T2 1 + (1.30e3 + 7.38e3i)T + (-1.26e9 + 4.61e8i)T^{2}
71 1+(−2.77e4+1.01e4i)T+(1.38e9−1.15e9i)T2 1 + (-2.77e4 + 1.01e4i)T + (1.38e9 - 1.15e9i)T^{2}
73 1+(−2.47e4−2.07e4i)T+(3.59e8+2.04e9i)T2 1 + (-2.47e4 - 2.07e4i)T + (3.59e8 + 2.04e9i)T^{2}
79 1+(2.36e4+1.98e4i)T+(5.34e8+3.03e9i)T2 1 + (2.36e4 + 1.98e4i)T + (5.34e8 + 3.03e9i)T^{2}
83 1+(1.87e4+3.24e4i)T+(−1.96e9+3.41e9i)T2 1 + (1.87e4 + 3.24e4i)T + (-1.96e9 + 3.41e9i)T^{2}
89 1+(6.39e3−5.36e3i)T+(9.69e8−5.49e9i)T2 1 + (6.39e3 - 5.36e3i)T + (9.69e8 - 5.49e9i)T^{2}
97 1+(7.30e3−4.14e4i)T+(−8.06e9−2.93e9i)T2 1 + (7.30e3 - 4.14e4i)T + (-8.06e9 - 2.93e9i)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

−15.14708791397097383289005187128, −14.70113524142166235418536858991, −12.68178490041511549805409527525, −11.69427955081152224351426093407, −11.44590244848684811428634321675, −9.373321672842540095907134468666, −7.999555889357858115905935488301, −6.17584639200331232613518005680, −4.42075957654681948227761307313, −2.23723577448634154013816981948, 0.22627110144835400123007686983, 4.20829930971042469622553525549, 5.09696406972937024038050839794, 7.37235325014014421800265156863, 8.047100564611777405093799523771, 10.15989810538609631551824781702, 11.20543003286860670789131790870, 12.60970748509347213112356727689, 14.06087927583854463736932863295, 15.07780158184546330298368182796

Graph of the ZZ-function along the critical line