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Properties

Label 2-175-175.103-c1-0-1
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(103,⋅)\chi_{175} (103, \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

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

Graph of the ZZ-function along the critical line