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Properties

Label 2-175-175.17-c1-0-12
Degree 22
Conductor 175175
Sign −0.650+0.759i-0.650 + 0.759i
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  + (−1.61 + 0.0848i)2-s + (1.71 − 2.12i)3-s + (0.622 − 0.0653i)4-s + (−2.19 + 0.449i)5-s + (−2.60 + 3.58i)6-s + (−0.156 − 2.64i)7-s + (2.19 − 0.348i)8-s + (−0.927 − 4.36i)9-s + (3.50 − 0.912i)10-s + (−3.84 − 0.816i)11-s + (0.930 − 1.43i)12-s + (−0.0865 + 0.0441i)13-s + (0.476 + 4.26i)14-s + (−2.81 + 5.42i)15-s + (−4.75 + 1.01i)16-s + (−1.20 + 3.12i)17-s + ⋯
L(s)  = 1  + (−1.14 + 0.0599i)2-s + (0.992 − 1.22i)3-s + (0.311 − 0.0326i)4-s + (−0.979 + 0.200i)5-s + (−1.06 + 1.46i)6-s + (−0.0590 − 0.998i)7-s + (0.777 − 0.123i)8-s + (−0.309 − 1.45i)9-s + (1.10 − 0.288i)10-s + (−1.15 − 0.246i)11-s + (0.268 − 0.413i)12-s + (−0.0240 + 0.0122i)13-s + (0.127 + 1.13i)14-s + (−0.726 + 1.40i)15-s + (−1.18 + 0.252i)16-s + (−0.291 + 0.758i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.650+0.759i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((−0.650+0.759i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.650+0.759i-0.650 + 0.759i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(17,⋅)\chi_{175} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), −0.650+0.759i)(2,\ 175,\ (\ :1/2),\ -0.650 + 0.759i)

Particular Values

L(1)L(1) ≈\approx 0.245892−0.534418i0.245892 - 0.534418i
L(12)L(\frac12) ≈\approx 0.245892−0.534418i0.245892 - 0.534418i
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+(2.19−0.449i)T 1 + (2.19 - 0.449i)T
7 1+(0.156+2.64i)T 1 + (0.156 + 2.64i)T
good2 1+(1.61−0.0848i)T+(1.98−0.209i)T2 1 + (1.61 - 0.0848i)T + (1.98 - 0.209i)T^{2}
3 1+(−1.71+2.12i)T+(−0.623−2.93i)T2 1 + (-1.71 + 2.12i)T + (-0.623 - 2.93i)T^{2}
11 1+(3.84+0.816i)T+(10.0+4.47i)T2 1 + (3.84 + 0.816i)T + (10.0 + 4.47i)T^{2}
13 1+(0.0865−0.0441i)T+(7.64−10.5i)T2 1 + (0.0865 - 0.0441i)T + (7.64 - 10.5i)T^{2}
17 1+(1.20−3.12i)T+(−12.6−11.3i)T2 1 + (1.20 - 3.12i)T + (-12.6 - 11.3i)T^{2}
19 1+(−0.637+6.06i)T+(−18.5−3.95i)T2 1 + (-0.637 + 6.06i)T + (-18.5 - 3.95i)T^{2}
23 1+(0.274+5.23i)T+(−22.8+2.40i)T2 1 + (0.274 + 5.23i)T + (-22.8 + 2.40i)T^{2}
29 1+(−2.51−3.45i)T+(−8.96+27.5i)T2 1 + (-2.51 - 3.45i)T + (-8.96 + 27.5i)T^{2}
31 1+(0.287+0.644i)T+(−20.7+23.0i)T2 1 + (0.287 + 0.644i)T + (-20.7 + 23.0i)T^{2}
37 1+(−6.47−4.20i)T+(15.0+33.8i)T2 1 + (-6.47 - 4.20i)T + (15.0 + 33.8i)T^{2}
41 1+(−0.124−0.0405i)T+(33.1+24.0i)T2 1 + (-0.124 - 0.0405i)T + (33.1 + 24.0i)T^{2}
43 1+(−8.25+8.25i)T−43iT2 1 + (-8.25 + 8.25i)T - 43iT^{2}
47 1+(−2.52+0.970i)T+(34.9−31.4i)T2 1 + (-2.52 + 0.970i)T + (34.9 - 31.4i)T^{2}
53 1+(−9.13−7.40i)T+(11.0+51.8i)T2 1 + (-9.13 - 7.40i)T + (11.0 + 51.8i)T^{2}
59 1+(3.45+3.83i)T+(−6.16+58.6i)T2 1 + (3.45 + 3.83i)T + (-6.16 + 58.6i)T^{2}
61 1+(−5.30−4.77i)T+(6.37+60.6i)T2 1 + (-5.30 - 4.77i)T + (6.37 + 60.6i)T^{2}
67 1+(0.521+0.200i)T+(49.7+44.8i)T2 1 + (0.521 + 0.200i)T + (49.7 + 44.8i)T^{2}
71 1+(−0.818+0.594i)T+(21.9−67.5i)T2 1 + (-0.818 + 0.594i)T + (21.9 - 67.5i)T^{2}
73 1+(1.16+1.78i)T+(−29.6+66.6i)T2 1 + (1.16 + 1.78i)T + (-29.6 + 66.6i)T^{2}
79 1+(6.28−14.1i)T+(−52.8−58.7i)T2 1 + (6.28 - 14.1i)T + (-52.8 - 58.7i)T^{2}
83 1+(1.69+10.6i)T+(−78.9+25.6i)T2 1 + (1.69 + 10.6i)T + (-78.9 + 25.6i)T^{2}
89 1+(0.671−0.745i)T+(−9.30−88.5i)T2 1 + (0.671 - 0.745i)T + (-9.30 - 88.5i)T^{2}
97 1+(−1.77+11.1i)T+(−92.2−29.9i)T2 1 + (-1.77 + 11.1i)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

−12.60053693523658359875899137985, −11.06022124121242538530456019794, −10.35100931284804131817473100001, −8.834994876578255259352785140992, −8.225944984093883477973071196389, −7.43454353063621751289141090794, −6.87190228515827188607691936518, −4.33175903075558180590606434754, −2.68702647272042775487611078696, −0.71544412926689194672832178930, 2.64316838740287633915573303706, 4.11155813620891829886366045723, 5.25428453878943160595979614933, 7.69384260101516957106440280985, 8.165093970987205626388230943505, 9.142098206965386804367483985567, 9.746441421036981997590479370168, 10.70004848626941954287720432121, 11.76120108834835600436593433991, 13.08092034749992921466873928581

Graph of the ZZ-function along the critical line