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Properties

Label 2-171-19.7-c3-0-7
Degree 22
Conductor 171171
Sign −0.288−0.957i-0.288 - 0.957i
Analytic cond. 10.089310.0893
Root an. cond. 3.176373.17637
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  + (1.45 + 2.52i)2-s + (−0.241 + 0.419i)4-s + (10.1 + 17.4i)5-s − 1.64·7-s + 21.8·8-s + (−29.4 + 50.9i)10-s − 8.92·11-s + (19.6 − 34.0i)13-s + (−2.39 − 4.14i)14-s + (33.8 + 58.5i)16-s + (3.95 + 6.84i)17-s + (−42.6 + 70.9i)19-s − 9.77·20-s + (−12.9 − 22.5i)22-s + (−55.2 + 95.7i)23-s + ⋯
L(s)  = 1  + (0.514 + 0.891i)2-s + (−0.0302 + 0.0523i)4-s + (0.903 + 1.56i)5-s − 0.0887·7-s + 0.967·8-s + (−0.930 + 1.61i)10-s − 0.244·11-s + (0.419 − 0.726i)13-s + (−0.0456 − 0.0791i)14-s + (0.528 + 0.915i)16-s + (0.0563 + 0.0976i)17-s + (−0.515 + 0.856i)19-s − 0.109·20-s + (−0.125 − 0.218i)22-s + (−0.501 + 0.868i)23-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((−0.288−0.957i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+3/2) L(s)=((−0.288−0.957i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: −0.288−0.957i-0.288 - 0.957i
Analytic conductor: 10.089310.0893
Root analytic conductor: 3.176373.17637
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ171(64,⋅)\chi_{171} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 171, ( :3/2), −0.288−0.957i)(2,\ 171,\ (\ :3/2),\ -0.288 - 0.957i)

Particular Values

L(2)L(2) ≈\approx 1.63222+2.19705i1.63222 + 2.19705i
L(12)L(\frac12) ≈\approx 1.63222+2.19705i1.63222 + 2.19705i
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
19 1+(42.6−70.9i)T 1 + (42.6 - 70.9i)T
good2 1+(−1.45−2.52i)T+(−4+6.92i)T2 1 + (-1.45 - 2.52i)T + (-4 + 6.92i)T^{2}
5 1+(−10.1−17.4i)T+(−62.5+108.i)T2 1 + (-10.1 - 17.4i)T + (-62.5 + 108. i)T^{2}
7 1+1.64T+343T2 1 + 1.64T + 343T^{2}
11 1+8.92T+1.33e3T2 1 + 8.92T + 1.33e3T^{2}
13 1+(−19.6+34.0i)T+(−1.09e3−1.90e3i)T2 1 + (-19.6 + 34.0i)T + (-1.09e3 - 1.90e3i)T^{2}
17 1+(−3.95−6.84i)T+(−2.45e3+4.25e3i)T2 1 + (-3.95 - 6.84i)T + (-2.45e3 + 4.25e3i)T^{2}
23 1+(55.2−95.7i)T+(−6.08e3−1.05e4i)T2 1 + (55.2 - 95.7i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(−71.8+124.i)T+(−1.21e4−2.11e4i)T2 1 + (-71.8 + 124. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+57.1T+2.97e4T2 1 + 57.1T + 2.97e4T^{2}
37 1−297.T+5.06e4T2 1 - 297.T + 5.06e4T^{2}
41 1+(112.+194.i)T+(−3.44e4+5.96e4i)T2 1 + (112. + 194. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(163.+283.i)T+(−3.97e4+6.88e4i)T2 1 + (163. + 283. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−56.2+97.4i)T+(−5.19e4−8.99e4i)T2 1 + (-56.2 + 97.4i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−305.+529.i)T+(−7.44e4−1.28e5i)T2 1 + (-305. + 529. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(−368.−638.i)T+(−1.02e5+1.77e5i)T2 1 + (-368. - 638. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−420.+728.i)T+(−1.13e5−1.96e5i)T2 1 + (-420. + 728. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(53.2−92.1i)T+(−1.50e5−2.60e5i)T2 1 + (53.2 - 92.1i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+(−386.−669.i)T+(−1.78e5+3.09e5i)T2 1 + (-386. - 669. i)T + (-1.78e5 + 3.09e5i)T^{2}
73 1+(369.+639.i)T+(−1.94e5+3.36e5i)T2 1 + (369. + 639. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(257.+446.i)T+(−2.46e5+4.26e5i)T2 1 + (257. + 446. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−618.T+5.71e5T2 1 - 618.T + 5.71e5T^{2}
89 1+(−62.3+107.i)T+(−3.52e5−6.10e5i)T2 1 + (-62.3 + 107. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+(349.+606.i)T+(−4.56e5+7.90e5i)T2 1 + (349. + 606. i)T + (-4.56e5 + 7.90e5i)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

−13.10691425823830010260067581962, −11.42883590543871151825895217032, −10.40316450012201399803986231678, −9.955892046849952961847127280884, −8.081754865939062311707571231266, −7.05894533382096819218646152777, −6.15752427437959787819132434123, −5.51370703314719426412116286495, −3.64536772331161202930359791299, −2.08158222208045279979520750757, 1.20071454383631981979454390394, 2.45295767510329918899264673250, 4.25323613382788632076420033518, 5.04808308950224286868258194804, 6.46146070791523680206872270116, 8.143257611804119134967492012745, 9.093535756996221316614314868427, 10.07745622574242892543056138260, 11.21955036217177469241004836275, 12.19851932679573860905790537671

Graph of the ZZ-function along the critical line