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Properties

Label 2-175-35.4-c3-0-17
Degree 22
Conductor 175175
Sign 0.0841+0.996i0.0841 + 0.996i
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  + (−4.35 + 2.51i)2-s + (−7.22 − 4.17i)3-s + (8.62 − 14.9i)4-s + 41.9·6-s + (7.81 − 16.7i)7-s + 46.5i·8-s + (21.3 + 36.9i)9-s + (0.444 − 0.769i)11-s + (−124. + 71.9i)12-s + 25.9i·13-s + (8.15 + 92.7i)14-s + (−47.8 − 82.8i)16-s + (83.4 + 48.1i)17-s + (−185. − 107. i)18-s + (−44.5 − 77.1i)19-s + ⋯
L(s)  = 1  + (−1.53 + 0.888i)2-s + (−1.39 − 0.802i)3-s + (1.07 − 1.86i)4-s + 2.85·6-s + (0.422 − 0.906i)7-s + 2.05i·8-s + (0.789 + 1.36i)9-s + (0.0121 − 0.0210i)11-s + (−2.99 + 1.73i)12-s + 0.553i·13-s + (0.155 + 1.76i)14-s + (−0.747 − 1.29i)16-s + (1.18 + 0.687i)17-s + (−2.42 − 1.40i)18-s + (−0.537 − 0.931i)19-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.0841+0.996i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.0841+0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0841 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.0841+0.996i0.0841 + 0.996i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(74,⋅)\chi_{175} (74, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), 0.0841+0.996i)(2,\ 175,\ (\ :3/2),\ 0.0841 + 0.996i)

Particular Values

L(2)L(2) ≈\approx 0.284985−0.261927i0.284985 - 0.261927i
L(12)L(\frac12) ≈\approx 0.284985−0.261927i0.284985 - 0.261927i
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 1
7 1+(−7.81+16.7i)T 1 + (-7.81 + 16.7i)T
good2 1+(4.35−2.51i)T+(4−6.92i)T2 1 + (4.35 - 2.51i)T + (4 - 6.92i)T^{2}
3 1+(7.22+4.17i)T+(13.5+23.3i)T2 1 + (7.22 + 4.17i)T + (13.5 + 23.3i)T^{2}
11 1+(−0.444+0.769i)T+(−665.5−1.15e3i)T2 1 + (-0.444 + 0.769i)T + (-665.5 - 1.15e3i)T^{2}
13 1−25.9iT−2.19e3T2 1 - 25.9iT - 2.19e3T^{2}
17 1+(−83.4−48.1i)T+(2.45e3+4.25e3i)T2 1 + (-83.4 - 48.1i)T + (2.45e3 + 4.25e3i)T^{2}
19 1+(44.5+77.1i)T+(−3.42e3+5.94e3i)T2 1 + (44.5 + 77.1i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(−100.+58.2i)T+(6.08e3−1.05e4i)T2 1 + (-100. + 58.2i)T + (6.08e3 - 1.05e4i)T^{2}
29 1−222.T+2.43e4T2 1 - 222.T + 2.43e4T^{2}
31 1+(−6.45+11.1i)T+(−1.48e4−2.57e4i)T2 1 + (-6.45 + 11.1i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(−79.0+45.6i)T+(2.53e4−4.38e4i)T2 1 + (-79.0 + 45.6i)T + (2.53e4 - 4.38e4i)T^{2}
41 1+98.4T+6.89e4T2 1 + 98.4T + 6.89e4T^{2}
43 1+392.iT−7.95e4T2 1 + 392. iT - 7.95e4T^{2}
47 1+(190.−110.i)T+(5.19e4−8.99e4i)T2 1 + (190. - 110. i)T + (5.19e4 - 8.99e4i)T^{2}
53 1+(198.+114.i)T+(7.44e4+1.28e5i)T2 1 + (198. + 114. i)T + (7.44e4 + 1.28e5i)T^{2}
59 1+(−6.76+11.7i)T+(−1.02e5−1.77e5i)T2 1 + (-6.76 + 11.7i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(102.+178.i)T+(−1.13e5+1.96e5i)T2 1 + (102. + 178. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(282.+162.i)T+(1.50e5+2.60e5i)T2 1 + (282. + 162. i)T + (1.50e5 + 2.60e5i)T^{2}
71 1+583.T+3.57e5T2 1 + 583.T + 3.57e5T^{2}
73 1+(823.+475.i)T+(1.94e5+3.36e5i)T2 1 + (823. + 475. i)T + (1.94e5 + 3.36e5i)T^{2}
79 1+(−225.−390.i)T+(−2.46e5+4.26e5i)T2 1 + (-225. - 390. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1+164.iT−5.71e5T2 1 + 164. iT - 5.71e5T^{2}
89 1+(−442.−766.i)T+(−3.52e5+6.10e5i)T2 1 + (-442. - 766. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−62.1iT−9.12e5T2 1 - 62.1iT - 9.12e5T^{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

−11.64459358483245686670225629337, −10.77505920984269955131758489447, −10.17821432633369706685866686519, −8.727833940331947544000708824422, −7.65305452151072126340431801700, −6.88474872694296039142112499195, −6.18391013262816311297379030208, −4.88904828302003257623634326208, −1.44922318786353931893958605411, −0.45172628101231704806514469830, 1.14196292134310775067671760250, 3.05556374144475994004860024213, 4.91821747766078774047149817629, 6.08836786819580304643390178572, 7.68450973847761111023681387073, 8.760746773047692545320799849929, 9.856993838817028459367233246799, 10.36785211604148881299407159564, 11.39286530183300659128128219470, 11.87446230865902794229961742473

Graph of the ZZ-function along the critical line