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Properties

Label 2-175-7.2-c3-0-21
Degree 22
Conductor 175175
Sign 0.997−0.0706i0.997 - 0.0706i
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
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  + (−2.20 + 3.82i)2-s + (1.62 + 2.80i)3-s + (−5.74 − 9.94i)4-s − 14.3·6-s + (−16.1 − 9.14i)7-s + 15.3·8-s + (8.24 − 14.2i)9-s + (0.0710 + 0.123i)11-s + (18.6 − 32.2i)12-s + 32.1·13-s + (70.4 − 41.3i)14-s + (11.9 − 20.7i)16-s + (−57.1 − 99.0i)17-s + (36.3 + 63.0i)18-s + (−21.6 + 37.4i)19-s + ⋯
L(s)  = 1  + (−0.780 + 1.35i)2-s + (0.312 + 0.540i)3-s + (−0.717 − 1.24i)4-s − 0.973·6-s + (−0.869 − 0.493i)7-s + 0.679·8-s + (0.305 − 0.528i)9-s + (0.00194 + 0.00337i)11-s + (0.447 − 0.775i)12-s + 0.685·13-s + (1.34 − 0.790i)14-s + (0.187 − 0.324i)16-s + (−0.815 − 1.41i)17-s + (0.476 + 0.825i)18-s + (−0.260 + 0.452i)19-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.997−0.0706i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.997−0.0706i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.997−0.0706i0.997 - 0.0706i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(51,⋅)\chi_{175} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), 0.997−0.0706i)(2,\ 175,\ (\ :3/2),\ 0.997 - 0.0706i)

Particular Values

L(2)L(2) ≈\approx 0.746826+0.0264190i0.746826 + 0.0264190i
L(12)L(\frac12) ≈\approx 0.746826+0.0264190i0.746826 + 0.0264190i
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)
bad5 1 1
7 1+(16.1+9.14i)T 1 + (16.1 + 9.14i)T
good2 1+(2.20−3.82i)T+(−4−6.92i)T2 1 + (2.20 - 3.82i)T + (-4 - 6.92i)T^{2}
3 1+(−1.62−2.80i)T+(−13.5+23.3i)T2 1 + (-1.62 - 2.80i)T + (-13.5 + 23.3i)T^{2}
11 1+(−0.0710−0.123i)T+(−665.5+1.15e3i)T2 1 + (-0.0710 - 0.123i)T + (-665.5 + 1.15e3i)T^{2}
13 1−32.1T+2.19e3T2 1 - 32.1T + 2.19e3T^{2}
17 1+(57.1+99.0i)T+(−2.45e3+4.25e3i)T2 1 + (57.1 + 99.0i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(21.6−37.4i)T+(−3.42e3−5.94e3i)T2 1 + (21.6 - 37.4i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−77.2+133.i)T+(−6.08e3−1.05e4i)T2 1 + (-77.2 + 133. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+40.1T+2.43e4T2 1 + 40.1T + 2.43e4T^{2}
31 1+(37.7+65.3i)T+(−1.48e4+2.57e4i)T2 1 + (37.7 + 65.3i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(200.−346.i)T+(−2.53e4−4.38e4i)T2 1 + (200. - 346. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+95.4T+6.89e4T2 1 + 95.4T + 6.89e4T^{2}
43 1−340.T+7.95e4T2 1 - 340.T + 7.95e4T^{2}
47 1+(3.74−6.48i)T+(−5.19e4−8.99e4i)T2 1 + (3.74 - 6.48i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(338.+586.i)T+(−7.44e4+1.28e5i)T2 1 + (338. + 586. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(398.+689.i)T+(−1.02e5+1.77e5i)T2 1 + (398. + 689. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−378.+655.i)T+(−1.13e5−1.96e5i)T2 1 + (-378. + 655. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(370.+641.i)T+(−1.50e5+2.60e5i)T2 1 + (370. + 641. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−37.0T+3.57e5T2 1 - 37.0T + 3.57e5T^{2}
73 1+(−40.4−70.0i)T+(−1.94e5+3.36e5i)T2 1 + (-40.4 - 70.0i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(−158.+274.i)T+(−2.46e5−4.26e5i)T2 1 + (-158. + 274. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−945.T+5.71e5T2 1 - 945.T + 5.71e5T^{2}
89 1+(391.−678.i)T+(−3.52e5−6.10e5i)T2 1 + (391. - 678. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+393.T+9.12e5T2 1 + 393.T + 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

−12.40829537592478229807864523193, −10.85747441423551795243915261607, −9.737897684390995486412421068998, −9.201739799802816013860575091865, −8.219331503408069841046718738193, −6.88231215486654320412216756869, −6.41686348523769278359931107525, −4.79282411161444895706586429762, −3.31415683198840291245710616870, −0.44381878714096277187893656264, 1.46230159442991157565174763748, 2.60273616426828695698773556412, 3.87502284639498457921377694002, 5.94100655226129177340568679603, 7.31475374520168267770387888868, 8.651923546823838100460404959365, 9.155430338939792693972064695472, 10.45079361965578099777883756069, 11.01653833421437204358340154113, 12.27583347986876047191124748397

Graph of the ZZ-function along the critical line