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Properties

Label 2-175-175.103-c1-0-7
Degree 22
Conductor 175175
Sign 0.953−0.302i0.953 - 0.302i
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  + (−1.80 − 0.0947i)2-s + (1.32 + 1.64i)3-s + (1.27 + 0.133i)4-s + (2.03 − 0.932i)5-s + (−2.24 − 3.09i)6-s + (0.405 − 2.61i)7-s + (1.29 + 0.204i)8-s + (−0.303 + 1.42i)9-s + (−3.76 + 1.49i)10-s + (−2.23 + 0.474i)11-s + (1.46 + 2.26i)12-s + (4.21 + 2.14i)13-s + (−0.981 + 4.68i)14-s + (4.23 + 2.09i)15-s + (−4.81 − 1.02i)16-s + (1.33 + 3.47i)17-s + ⋯
L(s)  = 1  + (−1.27 − 0.0669i)2-s + (0.767 + 0.947i)3-s + (0.635 + 0.0667i)4-s + (0.908 − 0.417i)5-s + (−0.917 − 1.26i)6-s + (0.153 − 0.988i)7-s + (0.456 + 0.0723i)8-s + (−0.101 + 0.475i)9-s + (−1.18 + 0.472i)10-s + (−0.672 + 0.142i)11-s + (0.424 + 0.653i)12-s + (1.16 + 0.595i)13-s + (−0.262 + 1.25i)14-s + (1.09 + 0.541i)15-s + (−1.20 − 0.255i)16-s + (0.323 + 0.841i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.953−0.302i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.953−0.302i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.953−0.302i0.953 - 0.302i
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), 0.953−0.302i)(2,\ 175,\ (\ :1/2),\ 0.953 - 0.302i)

Particular Values

L(1)L(1) ≈\approx 0.887350+0.137579i0.887350 + 0.137579i
L(12)L(\frac12) ≈\approx 0.887350+0.137579i0.887350 + 0.137579i
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+(−2.03+0.932i)T 1 + (-2.03 + 0.932i)T
7 1+(−0.405+2.61i)T 1 + (-0.405 + 2.61i)T
good2 1+(1.80+0.0947i)T+(1.98+0.209i)T2 1 + (1.80 + 0.0947i)T + (1.98 + 0.209i)T^{2}
3 1+(−1.32−1.64i)T+(−0.623+2.93i)T2 1 + (-1.32 - 1.64i)T + (-0.623 + 2.93i)T^{2}
11 1+(2.23−0.474i)T+(10.0−4.47i)T2 1 + (2.23 - 0.474i)T + (10.0 - 4.47i)T^{2}
13 1+(−4.21−2.14i)T+(7.64+10.5i)T2 1 + (-4.21 - 2.14i)T + (7.64 + 10.5i)T^{2}
17 1+(−1.33−3.47i)T+(−12.6+11.3i)T2 1 + (-1.33 - 3.47i)T + (-12.6 + 11.3i)T^{2}
19 1+(0.216+2.05i)T+(−18.5+3.95i)T2 1 + (0.216 + 2.05i)T + (-18.5 + 3.95i)T^{2}
23 1+(0.0829−1.58i)T+(−22.8−2.40i)T2 1 + (0.0829 - 1.58i)T + (-22.8 - 2.40i)T^{2}
29 1+(6.19−8.53i)T+(−8.96−27.5i)T2 1 + (6.19 - 8.53i)T + (-8.96 - 27.5i)T^{2}
31 1+(−1.38+3.11i)T+(−20.7−23.0i)T2 1 + (-1.38 + 3.11i)T + (-20.7 - 23.0i)T^{2}
37 1+(1.80−1.17i)T+(15.0−33.8i)T2 1 + (1.80 - 1.17i)T + (15.0 - 33.8i)T^{2}
41 1+(4.99−1.62i)T+(33.1−24.0i)T2 1 + (4.99 - 1.62i)T + (33.1 - 24.0i)T^{2}
43 1+(6.27+6.27i)T+43iT2 1 + (6.27 + 6.27i)T + 43iT^{2}
47 1+(6.00+2.30i)T+(34.9+31.4i)T2 1 + (6.00 + 2.30i)T + (34.9 + 31.4i)T^{2}
53 1+(3.15−2.55i)T+(11.0−51.8i)T2 1 + (3.15 - 2.55i)T + (11.0 - 51.8i)T^{2}
59 1+(−8.50+9.44i)T+(−6.16−58.6i)T2 1 + (-8.50 + 9.44i)T + (-6.16 - 58.6i)T^{2}
61 1+(0.722−0.650i)T+(6.37−60.6i)T2 1 + (0.722 - 0.650i)T + (6.37 - 60.6i)T^{2}
67 1+(−9.57+3.67i)T+(49.7−44.8i)T2 1 + (-9.57 + 3.67i)T + (49.7 - 44.8i)T^{2}
71 1+(1.86+1.35i)T+(21.9+67.5i)T2 1 + (1.86 + 1.35i)T + (21.9 + 67.5i)T^{2}
73 1+(4.27−6.58i)T+(−29.6−66.6i)T2 1 + (4.27 - 6.58i)T + (-29.6 - 66.6i)T^{2}
79 1+(4.69+10.5i)T+(−52.8+58.7i)T2 1 + (4.69 + 10.5i)T + (-52.8 + 58.7i)T^{2}
83 1+(0.909−5.73i)T+(−78.9−25.6i)T2 1 + (0.909 - 5.73i)T + (-78.9 - 25.6i)T^{2}
89 1+(4.47+4.96i)T+(−9.30+88.5i)T2 1 + (4.47 + 4.96i)T + (-9.30 + 88.5i)T^{2}
97 1+(−0.453−2.86i)T+(−92.2+29.9i)T2 1 + (-0.453 - 2.86i)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

−13.07814255627351701961453353091, −11.11053263854284980874143142846, −10.33917675888792270995233323508, −9.733507759151260286057643259769, −8.841236619389063351066438995784, −8.193997597528185509004617275988, −6.78843173216019079219414179524, −4.99750786637677396197013132397, −3.69830820686217174089275104814, −1.61965023190202143339984723878, 1.65520564261159025494348415708, 2.80935563889523435262909064245, 5.49438649011679169972901056546, 6.71847062583564697538434948589, 7.931183046560204477808784710717, 8.452912396832560734833371872007, 9.454986886006782610416006463353, 10.36012307199290581393189061146, 11.45803625846814305770474350609, 12.98898226359515477803640489384

Graph of the ZZ-function along the critical line