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Properties

Label 2-175-175.103-c3-0-21
Degree 22
Conductor 175175
Sign −0.228+0.973i-0.228 + 0.973i
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  + (−3.05 − 0.160i)2-s + (−5.77 − 7.12i)3-s + (1.35 + 0.142i)4-s + (−5.21 + 9.88i)5-s + (16.4 + 22.7i)6-s + (7.73 − 16.8i)7-s + (20.0 + 3.17i)8-s + (−11.8 + 55.8i)9-s + (17.5 − 29.3i)10-s + (26.7 − 5.67i)11-s + (−6.81 − 10.4i)12-s + (30.7 + 15.6i)13-s + (−26.3 + 50.1i)14-s + (100. − 19.8i)15-s + (−71.4 − 15.1i)16-s + (18.9 + 49.4i)17-s + ⋯
L(s)  = 1  + (−1.08 − 0.0566i)2-s + (−1.11 − 1.37i)3-s + (0.169 + 0.0178i)4-s + (−0.466 + 0.884i)5-s + (1.12 + 1.54i)6-s + (0.417 − 0.908i)7-s + (0.886 + 0.140i)8-s + (−0.439 + 2.06i)9-s + (0.554 − 0.929i)10-s + (0.732 − 0.155i)11-s + (−0.163 − 0.252i)12-s + (0.656 + 0.334i)13-s + (−0.502 + 0.957i)14-s + (1.73 − 0.342i)15-s + (−1.11 − 0.237i)16-s + (0.270 + 0.705i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.228+0.973i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.228+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.228+0.973i-0.228 + 0.973i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(103,⋅)\chi_{175} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), −0.228+0.973i)(2,\ 175,\ (\ :3/2),\ -0.228 + 0.973i)

Particular Values

L(2)L(2) ≈\approx 0.311973−0.393533i0.311973 - 0.393533i
L(12)L(\frac12) ≈\approx 0.311973−0.393533i0.311973 - 0.393533i
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+(5.21−9.88i)T 1 + (5.21 - 9.88i)T
7 1+(−7.73+16.8i)T 1 + (-7.73 + 16.8i)T
good2 1+(3.05+0.160i)T+(7.95+0.836i)T2 1 + (3.05 + 0.160i)T + (7.95 + 0.836i)T^{2}
3 1+(5.77+7.12i)T+(−5.61+26.4i)T2 1 + (5.77 + 7.12i)T + (-5.61 + 26.4i)T^{2}
11 1+(−26.7+5.67i)T+(1.21e3−541.i)T2 1 + (-26.7 + 5.67i)T + (1.21e3 - 541. i)T^{2}
13 1+(−30.7−15.6i)T+(1.29e3+1.77e3i)T2 1 + (-30.7 - 15.6i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−18.9−49.4i)T+(−3.65e3+3.28e3i)T2 1 + (-18.9 - 49.4i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−1.99−18.9i)T+(−6.70e3+1.42e3i)T2 1 + (-1.99 - 18.9i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−5.72+109.i)T+(−1.21e4−1.27e3i)T2 1 + (-5.72 + 109. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(142.−195.i)T+(−7.53e3−2.31e4i)T2 1 + (142. - 195. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−45.9+103.i)T+(−1.99e4−2.21e4i)T2 1 + (-45.9 + 103. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−146.+95.2i)T+(2.06e4−4.62e4i)T2 1 + (-146. + 95.2i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−190.+62.0i)T+(5.57e4−4.05e4i)T2 1 + (-190. + 62.0i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(131.+131.i)T+7.95e4iT2 1 + (131. + 131. i)T + 7.95e4iT^{2}
47 1+(−528.−202.i)T+(7.71e4+6.94e4i)T2 1 + (-528. - 202. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(−229.+185.i)T+(3.09e4−1.45e5i)T2 1 + (-229. + 185. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−290.+322.i)T+(−2.14e4−2.04e5i)T2 1 + (-290. + 322. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(50.4−45.4i)T+(2.37e4−2.25e5i)T2 1 + (50.4 - 45.4i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(714.−274.i)T+(2.23e5−2.01e5i)T2 1 + (714. - 274. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(785.+570.i)T+(1.10e5+3.40e5i)T2 1 + (785. + 570. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(294.−453.i)T+(−1.58e5−3.55e5i)T2 1 + (294. - 453. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(−289.−650.i)T+(−3.29e5+3.66e5i)T2 1 + (-289. - 650. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−185.+1.17e3i)T+(−5.43e5−1.76e5i)T2 1 + (-185. + 1.17e3i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−560.−622.i)T+(−7.36e4+7.01e5i)T2 1 + (-560. - 622. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(223.+1.40e3i)T+(−8.68e5+2.82e5i)T2 1 + (223. + 1.40e3i)T + (-8.68e5 + 2.82e5i)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

−11.60902209611710157390517251510, −10.96089828261446936188156410444, −10.35801279968474436237795378865, −8.649183865712433303890623857548, −7.60676527939231334822633168683, −7.04517517413255515053064226435, −6.01303071352815608295236022855, −4.17736095828441301119849252325, −1.68280020015112242628458932690, −0.56223152848605722730800776435, 0.947326209706983411125694190355, 3.98189793627312422619762243252, 4.92921604000620666311505763642, 5.89349013270426208727538948603, 7.72165218974388184020034865983, 9.020534860027581581209959534226, 9.282094891061478872228082564867, 10.38893638677863372299388354816, 11.52956416628500044278422178298, 11.85663878666490521008145081166

Graph of the ZZ-function along the critical line