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Properties

Label 2-175-175.103-c1-0-6
Degree 22
Conductor 175175
Sign 0.716+0.697i0.716 + 0.697i
Analytic cond. 1.397381.39738
Root an. cond. 1.182101.18210
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.0642i)2-s + (0.158 + 0.195i)3-s + (−0.490 − 0.0515i)4-s + (−2.14 + 0.624i)5-s + (−0.181 − 0.249i)6-s + (2.13 − 1.56i)7-s + (3.02 + 0.478i)8-s + (0.610 − 2.87i)9-s + (2.67 − 0.627i)10-s + (6.21 − 1.32i)11-s + (−0.0675 − 0.104i)12-s + (−3.08 − 1.57i)13-s + (−2.71 + 1.77i)14-s + (−0.462 − 0.321i)15-s + (−2.71 − 0.576i)16-s + (1.38 + 3.61i)17-s + ⋯
L(s)  = 1  + (−0.866 − 0.0454i)2-s + (0.0914 + 0.112i)3-s + (−0.245 − 0.0257i)4-s + (−0.960 + 0.279i)5-s + (−0.0741 − 0.102i)6-s + (0.806 − 0.590i)7-s + (1.06 + 0.169i)8-s + (0.203 − 0.957i)9-s + (0.845 − 0.198i)10-s + (1.87 − 0.398i)11-s + (−0.0195 − 0.0300i)12-s + (−0.856 − 0.436i)13-s + (−0.726 + 0.475i)14-s + (−0.119 − 0.0828i)15-s + (−0.677 − 0.144i)16-s + (0.336 + 0.875i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.716+0.697i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.716+0.697i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.716+0.697i0.716 + 0.697i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(103,⋅)\chi_{175} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), 0.716+0.697i)(2,\ 175,\ (\ :1/2),\ 0.716 + 0.697i)

Particular Values

L(1)L(1) ≈\approx 0.623413−0.253186i0.623413 - 0.253186i
L(12)L(\frac12) ≈\approx 0.623413−0.253186i0.623413 - 0.253186i
L(32)L(\frac{3}{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+(2.14−0.624i)T 1 + (2.14 - 0.624i)T
7 1+(−2.13+1.56i)T 1 + (-2.13 + 1.56i)T
good2 1+(1.22+0.0642i)T+(1.98+0.209i)T2 1 + (1.22 + 0.0642i)T + (1.98 + 0.209i)T^{2}
3 1+(−0.158−0.195i)T+(−0.623+2.93i)T2 1 + (-0.158 - 0.195i)T + (-0.623 + 2.93i)T^{2}
11 1+(−6.21+1.32i)T+(10.0−4.47i)T2 1 + (-6.21 + 1.32i)T + (10.0 - 4.47i)T^{2}
13 1+(3.08+1.57i)T+(7.64+10.5i)T2 1 + (3.08 + 1.57i)T + (7.64 + 10.5i)T^{2}
17 1+(−1.38−3.61i)T+(−12.6+11.3i)T2 1 + (-1.38 - 3.61i)T + (-12.6 + 11.3i)T^{2}
19 1+(0.630+6.00i)T+(−18.5+3.95i)T2 1 + (0.630 + 6.00i)T + (-18.5 + 3.95i)T^{2}
23 1+(0.0931−1.77i)T+(−22.8−2.40i)T2 1 + (0.0931 - 1.77i)T + (-22.8 - 2.40i)T^{2}
29 1+(−1.19+1.63i)T+(−8.96−27.5i)T2 1 + (-1.19 + 1.63i)T + (-8.96 - 27.5i)T^{2}
31 1+(−0.613+1.37i)T+(−20.7−23.0i)T2 1 + (-0.613 + 1.37i)T + (-20.7 - 23.0i)T^{2}
37 1+(6.46−4.19i)T+(15.0−33.8i)T2 1 + (6.46 - 4.19i)T + (15.0 - 33.8i)T^{2}
41 1+(−2.44+0.795i)T+(33.1−24.0i)T2 1 + (-2.44 + 0.795i)T + (33.1 - 24.0i)T^{2}
43 1+(−4.23−4.23i)T+43iT2 1 + (-4.23 - 4.23i)T + 43iT^{2}
47 1+(2.45+0.941i)T+(34.9+31.4i)T2 1 + (2.45 + 0.941i)T + (34.9 + 31.4i)T^{2}
53 1+(−2.37+1.91i)T+(11.0−51.8i)T2 1 + (-2.37 + 1.91i)T + (11.0 - 51.8i)T^{2}
59 1+(8.64−9.59i)T+(−6.16−58.6i)T2 1 + (8.64 - 9.59i)T + (-6.16 - 58.6i)T^{2}
61 1+(0.0639−0.0575i)T+(6.37−60.6i)T2 1 + (0.0639 - 0.0575i)T + (6.37 - 60.6i)T^{2}
67 1+(1.23−0.475i)T+(49.7−44.8i)T2 1 + (1.23 - 0.475i)T + (49.7 - 44.8i)T^{2}
71 1+(−1.14−0.833i)T+(21.9+67.5i)T2 1 + (-1.14 - 0.833i)T + (21.9 + 67.5i)T^{2}
73 1+(−7.23+11.1i)T+(−29.6−66.6i)T2 1 + (-7.23 + 11.1i)T + (-29.6 - 66.6i)T^{2}
79 1+(−1.79−4.03i)T+(−52.8+58.7i)T2 1 + (-1.79 - 4.03i)T + (-52.8 + 58.7i)T^{2}
83 1+(−0.0358+0.226i)T+(−78.9−25.6i)T2 1 + (-0.0358 + 0.226i)T + (-78.9 - 25.6i)T^{2}
89 1+(−8.63−9.59i)T+(−9.30+88.5i)T2 1 + (-8.63 - 9.59i)T + (-9.30 + 88.5i)T^{2}
97 1+(−0.0559−0.353i)T+(−92.2+29.9i)T2 1 + (-0.0559 - 0.353i)T + (-92.2 + 29.9i)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

−12.28018625301323317406708461383, −11.46229545911240961169533417458, −10.54775267805287994852209335954, −9.421270731277893155482004210324, −8.623531008189793896455749607390, −7.62446494634888174701025908356, −6.66729095789841649998106046342, −4.59604848344746306246005428209, −3.72126772053863369333731827380, −0.988986092425500048953726510190, 1.63845506802907923426566309168, 4.11782992855211580102522020655, 5.00874953678403521393653338676, 7.07924097901248467716583487478, 7.87485430651206341777300810742, 8.746331846680598457467746178273, 9.553148650646994500044868094296, 10.80305985826987898480929519419, 11.90945042496636084268823857501, 12.44475787895120183445323575712

Graph of the ZZ-function along the critical line