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Properties

Label 2-117-117.43-c3-0-2
Degree 22
Conductor 117117
Sign −0.741+0.671i-0.741 + 0.671i
Analytic cond. 6.903226.90322
Root an. cond. 2.627392.62739
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  + (−4.65 + 2.68i)2-s + (1.15 + 5.06i)3-s + (10.4 − 18.0i)4-s + (10.1 − 5.86i)5-s + (−18.9 − 20.4i)6-s − 8.42i·7-s + 68.9i·8-s + (−24.3 + 11.7i)9-s + (−31.5 + 54.5i)10-s + (−55.4 + 32.0i)11-s + (103. + 31.8i)12-s + (−38.7 − 26.4i)13-s + (22.6 + 39.1i)14-s + (41.4 + 44.7i)15-s + (−101. − 176. i)16-s + (30.6 + 53.0i)17-s + ⋯
L(s)  = 1  + (−1.64 + 0.949i)2-s + (0.222 + 0.974i)3-s + (1.30 − 2.25i)4-s + (0.909 − 0.524i)5-s + (−1.29 − 1.39i)6-s − 0.454i·7-s + 3.04i·8-s + (−0.900 + 0.434i)9-s + (−0.996 + 1.72i)10-s + (−1.52 + 0.877i)11-s + (2.48 + 0.767i)12-s + (−0.826 − 0.563i)13-s + (0.431 + 0.747i)14-s + (0.714 + 0.769i)15-s + (−1.58 − 2.75i)16-s + (0.436 + 0.756i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((−0.741+0.671i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+3/2) L(s)=((−0.741+0.671i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: −0.741+0.671i-0.741 + 0.671i
Analytic conductor: 6.903226.90322
Root analytic conductor: 2.627392.62739
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ117(43,⋅)\chi_{117} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :3/2), −0.741+0.671i)(2,\ 117,\ (\ :3/2),\ -0.741 + 0.671i)

Particular Values

L(2)L(2) ≈\approx 0.101113−0.262192i0.101113 - 0.262192i
L(12)L(\frac12) ≈\approx 0.101113−0.262192i0.101113 - 0.262192i
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)
bad3 1+(−1.15−5.06i)T 1 + (-1.15 - 5.06i)T
13 1+(38.7+26.4i)T 1 + (38.7 + 26.4i)T
good2 1+(4.65−2.68i)T+(4−6.92i)T2 1 + (4.65 - 2.68i)T + (4 - 6.92i)T^{2}
5 1+(−10.1+5.86i)T+(62.5−108.i)T2 1 + (-10.1 + 5.86i)T + (62.5 - 108. i)T^{2}
7 1+8.42iT−343T2 1 + 8.42iT - 343T^{2}
11 1+(55.4−32.0i)T+(665.5−1.15e3i)T2 1 + (55.4 - 32.0i)T + (665.5 - 1.15e3i)T^{2}
17 1+(−30.6−53.0i)T+(−2.45e3+4.25e3i)T2 1 + (-30.6 - 53.0i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(86.4−49.9i)T+(3.42e3−5.94e3i)T2 1 + (86.4 - 49.9i)T + (3.42e3 - 5.94e3i)T^{2}
23 1+106.T+1.21e4T2 1 + 106.T + 1.21e4T^{2}
29 1+(−72.3−125.i)T+(−1.21e4+2.11e4i)T2 1 + (-72.3 - 125. i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+(94.2−54.3i)T+(1.48e4−2.57e4i)T2 1 + (94.2 - 54.3i)T + (1.48e4 - 2.57e4i)T^{2}
37 1+(4.75+2.74i)T+(2.53e4+4.38e4i)T2 1 + (4.75 + 2.74i)T + (2.53e4 + 4.38e4i)T^{2}
41 1−66.6iT−6.89e4T2 1 - 66.6iT - 6.89e4T^{2}
43 1−331.T+7.95e4T2 1 - 331.T + 7.95e4T^{2}
47 1+(181.+104.i)T+(5.19e4+8.99e4i)T2 1 + (181. + 104. i)T + (5.19e4 + 8.99e4i)T^{2}
53 1+303.T+1.48e5T2 1 + 303.T + 1.48e5T^{2}
59 1+(576.+332.i)T+(1.02e5+1.77e5i)T2 1 + (576. + 332. i)T + (1.02e5 + 1.77e5i)T^{2}
61 1−636.T+2.26e5T2 1 - 636.T + 2.26e5T^{2}
67 1−138.iT−3.00e5T2 1 - 138. iT - 3.00e5T^{2}
71 1+(303.−175.i)T+(1.78e5−3.09e5i)T2 1 + (303. - 175. i)T + (1.78e5 - 3.09e5i)T^{2}
73 1+497.iT−3.89e5T2 1 + 497. iT - 3.89e5T^{2}
79 1+(1.08−1.87i)T+(−2.46e5−4.26e5i)T2 1 + (1.08 - 1.87i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+(−122.−70.8i)T+(2.85e5+4.95e5i)T2 1 + (-122. - 70.8i)T + (2.85e5 + 4.95e5i)T^{2}
89 1+(71.9+41.5i)T+(3.52e5+6.10e5i)T2 1 + (71.9 + 41.5i)T + (3.52e5 + 6.10e5i)T^{2}
97 1−510.iT−9.12e5T2 1 - 510. iT - 9.12e5T^{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

−14.24837799625505357806226848259, −12.70953000800045201468239199780, −10.59654234396214791272125831006, −10.25698743382883838839332583036, −9.528330716970586761839253574481, −8.368244534217071663748407993227, −7.58197968408878284870094761615, −5.91551836310522877792466778480, −5.00551599057266802427988604548, −2.08617030336363324387045663764, 0.22156887258503452094864377313, 2.21425833975579554618019564897, 2.71808599369814176031541099052, 6.05329047659187521770835720503, 7.34012588180400346186298325532, 8.225969374069199945754255820577, 9.253728250899668658568991242054, 10.22127629977443927855227194348, 11.19009276284356258798431461387, 12.16700655449680308005447550270

Graph of the ZZ-function along the critical line