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Properties

Label 2-88-11.5-c3-0-1
Degree 22
Conductor 8888
Sign 0.593−0.804i0.593 - 0.804i
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  + (−6.31 − 4.58i)3-s + (−3.65 + 11.2i)5-s + (7.70 − 5.60i)7-s + (10.4 + 32.2i)9-s + (30.3 + 20.2i)11-s + (20.4 + 63.0i)13-s + (74.7 − 54.3i)15-s + (2.53 − 7.79i)17-s + (88.2 + 64.1i)19-s − 74.3·21-s − 196.·23-s + (−12.3 − 8.94i)25-s + (16.7 − 51.6i)27-s + (−129. + 94.2i)29-s + (−25.0 − 77.2i)31-s + ⋯
L(s)  = 1  + (−1.21 − 0.883i)3-s + (−0.327 + 1.00i)5-s + (0.416 − 0.302i)7-s + (0.388 + 1.19i)9-s + (0.831 + 0.555i)11-s + (0.437 + 1.34i)13-s + (1.28 − 0.935i)15-s + (0.0361 − 0.111i)17-s + (1.06 + 0.774i)19-s − 0.773·21-s − 1.77·23-s + (−0.0984 − 0.0715i)25-s + (0.119 − 0.368i)27-s + (−0.831 + 0.603i)29-s + (−0.145 − 0.447i)31-s + ⋯

Functional equation

Λ(s)=(88s/2 ΓC(s) L(s)=((0.593−0.804i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(88s/2 ΓC(s+3/2) L(s)=((0.593−0.804i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8888    =    23⋅112^{3} \cdot 11
Sign: 0.593−0.804i0.593 - 0.804i
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), 0.593−0.804i)(2,\ 88,\ (\ :3/2),\ 0.593 - 0.804i)

Particular Values

L(2)L(2) ≈\approx 0.785702+0.396936i0.785702 + 0.396936i
L(12)L(\frac12) ≈\approx 0.785702+0.396936i0.785702 + 0.396936i
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+(−30.3−20.2i)T 1 + (-30.3 - 20.2i)T
good3 1+(6.31+4.58i)T+(8.34+25.6i)T2 1 + (6.31 + 4.58i)T + (8.34 + 25.6i)T^{2}
5 1+(3.65−11.2i)T+(−101.−73.4i)T2 1 + (3.65 - 11.2i)T + (-101. - 73.4i)T^{2}
7 1+(−7.70+5.60i)T+(105.−326.i)T2 1 + (-7.70 + 5.60i)T + (105. - 326. i)T^{2}
13 1+(−20.4−63.0i)T+(−1.77e3+1.29e3i)T2 1 + (-20.4 - 63.0i)T + (-1.77e3 + 1.29e3i)T^{2}
17 1+(−2.53+7.79i)T+(−3.97e3−2.88e3i)T2 1 + (-2.53 + 7.79i)T + (-3.97e3 - 2.88e3i)T^{2}
19 1+(−88.2−64.1i)T+(2.11e3+6.52e3i)T2 1 + (-88.2 - 64.1i)T + (2.11e3 + 6.52e3i)T^{2}
23 1+196.T+1.21e4T2 1 + 196.T + 1.21e4T^{2}
29 1+(129.−94.2i)T+(7.53e3−2.31e4i)T2 1 + (129. - 94.2i)T + (7.53e3 - 2.31e4i)T^{2}
31 1+(25.0+77.2i)T+(−2.41e4+1.75e4i)T2 1 + (25.0 + 77.2i)T + (-2.41e4 + 1.75e4i)T^{2}
37 1+(−94.2+68.4i)T+(1.56e4−4.81e4i)T2 1 + (-94.2 + 68.4i)T + (1.56e4 - 4.81e4i)T^{2}
41 1+(−289.−210.i)T+(2.12e4+6.55e4i)T2 1 + (-289. - 210. i)T + (2.12e4 + 6.55e4i)T^{2}
43 1−106.T+7.95e4T2 1 - 106.T + 7.95e4T^{2}
47 1+(210.+153.i)T+(3.20e4+9.87e4i)T2 1 + (210. + 153. i)T + (3.20e4 + 9.87e4i)T^{2}
53 1+(−170.−525.i)T+(−1.20e5+8.75e4i)T2 1 + (-170. - 525. i)T + (-1.20e5 + 8.75e4i)T^{2}
59 1+(−362.+263.i)T+(6.34e4−1.95e5i)T2 1 + (-362. + 263. i)T + (6.34e4 - 1.95e5i)T^{2}
61 1+(−165.+508.i)T+(−1.83e5−1.33e5i)T2 1 + (-165. + 508. i)T + (-1.83e5 - 1.33e5i)T^{2}
67 1+675.T+3.00e5T2 1 + 675.T + 3.00e5T^{2}
71 1+(234.−723.i)T+(−2.89e5−2.10e5i)T2 1 + (234. - 723. i)T + (-2.89e5 - 2.10e5i)T^{2}
73 1+(−500.+363.i)T+(1.20e5−3.69e5i)T2 1 + (-500. + 363. i)T + (1.20e5 - 3.69e5i)T^{2}
79 1+(−179.−551.i)T+(−3.98e5+2.89e5i)T2 1 + (-179. - 551. i)T + (-3.98e5 + 2.89e5i)T^{2}
83 1+(279.−860.i)T+(−4.62e5−3.36e5i)T2 1 + (279. - 860. i)T + (-4.62e5 - 3.36e5i)T^{2}
89 1−537.T+7.04e5T2 1 - 537.T + 7.04e5T^{2}
97 1+(574.+1.76e3i)T+(−7.38e5+5.36e5i)T2 1 + (574. + 1.76e3i)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

−13.93469464206975913917451624459, −12.43401640847337198395599394641, −11.58221545034552542307845208039, −11.07106830137574377775899099552, −9.626395458355299617913501092966, −7.66586249133520523669984043522, −6.86239574205685842785740037396, −5.90482945814154979325360232441, −4.07032565993840082637888949857, −1.58520732918764687842800214781, 0.67177888232135022234644451197, 3.92965712545785050405217067342, 5.17065304321924476415656176275, 5.97457182627543252887001647065, 8.005837841994473481454375527349, 9.181917234297547190619128728307, 10.37600826859315128691663360786, 11.49368410280977848252274019201, 12.05962272666478631006938372438, 13.33540589237471289997594638340

Graph of the ZZ-function along the critical line