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Properties

Label 2-88-11.5-c3-0-6
Degree 22
Conductor 8888
Sign 0.696+0.717i0.696 + 0.717i
Analytic cond. 5.192165.19216
Root an. cond. 2.278632.27863
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.61 + 1.89i)3-s + (6.32 − 19.4i)5-s + (−5.38 + 3.91i)7-s + (−5.12 − 15.7i)9-s + (25.0 − 26.5i)11-s + (16.3 + 50.2i)13-s + (53.4 − 38.8i)15-s + (18.3 − 56.5i)17-s + (89.7 + 65.1i)19-s − 21.4·21-s − 48.0·23-s + (−237. − 172. i)25-s + (43.4 − 133. i)27-s + (−100. + 72.8i)29-s + (94.2 + 290. i)31-s + ⋯
L(s)  = 1  + (0.502 + 0.365i)3-s + (0.565 − 1.74i)5-s + (−0.290 + 0.211i)7-s + (−0.189 − 0.583i)9-s + (0.685 − 0.728i)11-s + (0.348 + 1.07i)13-s + (0.920 − 0.668i)15-s + (0.262 − 0.807i)17-s + (1.08 + 0.787i)19-s − 0.223·21-s − 0.435·23-s + (−1.90 − 1.38i)25-s + (0.309 − 0.953i)27-s + (−0.642 + 0.466i)29-s + (0.546 + 1.68i)31-s + ⋯

Functional equation

Λ(s)=(88s/2 ΓC(s) L(s)=((0.696+0.717i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(88s/2 ΓC(s+3/2) L(s)=((0.696+0.717i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: 0.696+0.717i0.696 + 0.717i
Analytic conductor: 5.192165.19216
Root analytic conductor: 2.278632.27863
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ88(49,⋅)\chi_{88} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 88, ( :3/2), 0.696+0.717i)(2,\ 88,\ (\ :3/2),\ 0.696 + 0.717i)

Particular Values

L(2)L(2) ≈\approx 1.72877−0.731669i1.72877 - 0.731669i
L(12)L(\frac12) ≈\approx 1.72877−0.731669i1.72877 - 0.731669i
L(52)L(\frac{5}{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
11 1+(−25.0+26.5i)T 1 + (-25.0 + 26.5i)T
good3 1+(−2.61−1.89i)T+(8.34+25.6i)T2 1 + (-2.61 - 1.89i)T + (8.34 + 25.6i)T^{2}
5 1+(−6.32+19.4i)T+(−101.−73.4i)T2 1 + (-6.32 + 19.4i)T + (-101. - 73.4i)T^{2}
7 1+(5.38−3.91i)T+(105.−326.i)T2 1 + (5.38 - 3.91i)T + (105. - 326. i)T^{2}
13 1+(−16.3−50.2i)T+(−1.77e3+1.29e3i)T2 1 + (-16.3 - 50.2i)T + (-1.77e3 + 1.29e3i)T^{2}
17 1+(−18.3+56.5i)T+(−3.97e3−2.88e3i)T2 1 + (-18.3 + 56.5i)T + (-3.97e3 - 2.88e3i)T^{2}
19 1+(−89.7−65.1i)T+(2.11e3+6.52e3i)T2 1 + (-89.7 - 65.1i)T + (2.11e3 + 6.52e3i)T^{2}
23 1+48.0T+1.21e4T2 1 + 48.0T + 1.21e4T^{2}
29 1+(100.−72.8i)T+(7.53e3−2.31e4i)T2 1 + (100. - 72.8i)T + (7.53e3 - 2.31e4i)T^{2}
31 1+(−94.2−290.i)T+(−2.41e4+1.75e4i)T2 1 + (-94.2 - 290. i)T + (-2.41e4 + 1.75e4i)T^{2}
37 1+(128.−93.2i)T+(1.56e4−4.81e4i)T2 1 + (128. - 93.2i)T + (1.56e4 - 4.81e4i)T^{2}
41 1+(−184.−134.i)T+(2.12e4+6.55e4i)T2 1 + (-184. - 134. i)T + (2.12e4 + 6.55e4i)T^{2}
43 1+302.T+7.95e4T2 1 + 302.T + 7.95e4T^{2}
47 1+(0.196+0.142i)T+(3.20e4+9.87e4i)T2 1 + (0.196 + 0.142i)T + (3.20e4 + 9.87e4i)T^{2}
53 1+(41.2+127.i)T+(−1.20e5+8.75e4i)T2 1 + (41.2 + 127. i)T + (-1.20e5 + 8.75e4i)T^{2}
59 1+(227.−165.i)T+(6.34e4−1.95e5i)T2 1 + (227. - 165. i)T + (6.34e4 - 1.95e5i)T^{2}
61 1+(−165.+509.i)T+(−1.83e5−1.33e5i)T2 1 + (-165. + 509. i)T + (-1.83e5 - 1.33e5i)T^{2}
67 1−695.T+3.00e5T2 1 - 695.T + 3.00e5T^{2}
71 1+(118.−365.i)T+(−2.89e5−2.10e5i)T2 1 + (118. - 365. i)T + (-2.89e5 - 2.10e5i)T^{2}
73 1+(−616.+447.i)T+(1.20e5−3.69e5i)T2 1 + (-616. + 447. i)T + (1.20e5 - 3.69e5i)T^{2}
79 1+(71.6+220.i)T+(−3.98e5+2.89e5i)T2 1 + (71.6 + 220. i)T + (-3.98e5 + 2.89e5i)T^{2}
83 1+(232.−715.i)T+(−4.62e5−3.36e5i)T2 1 + (232. - 715. i)T + (-4.62e5 - 3.36e5i)T^{2}
89 1−1.19e3T+7.04e5T2 1 - 1.19e3T + 7.04e5T^{2}
97 1+(−20.9−64.4i)T+(−7.38e5+5.36e5i)T2 1 + (-20.9 - 64.4i)T + (-7.38e5 + 5.36e5i)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

−13.75188349945266998225829570537, −12.39622584825392848840094784187, −11.71285684289257954415221849538, −9.688479053128394346604318479017, −9.167555158951180392862646882100, −8.354370289888708530354989727082, −6.32367285345193611624001297397, −5.05446431333636937400022819176, −3.58618931059507346504485670572, −1.25109659199712826329364511889, 2.22126717945734530868991645545, 3.47151758470875997607260210357, 5.81116210793122571381739190435, 7.00147310330274712492604684628, 7.88213885627699402606765571388, 9.622548937219314714780555045846, 10.47381071635186547433024954126, 11.48043727510914162221714789082, 13.09580736465596685697589533463, 13.84719891030584087409180773818

Graph of the ZZ-function along the critical line