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Properties

Label 2-175-35.9-c3-0-24
Degree 22
Conductor 175175
Sign 0.315+0.949i0.315 + 0.949i
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  + (3.33 + 1.92i)2-s + (−7.76 + 4.48i)3-s + (3.39 + 5.88i)4-s − 34.4·6-s + (−13.8 − 12.3i)7-s − 4.64i·8-s + (26.7 − 46.2i)9-s + (−11.5 − 20.0i)11-s + (−52.7 − 30.4i)12-s + 61.0i·13-s + (−22.2 − 67.6i)14-s + (36.0 − 62.5i)16-s + (−0.596 + 0.344i)17-s + (177. − 102. i)18-s + (31.6 − 54.7i)19-s + ⋯
L(s)  = 1  + (1.17 + 0.679i)2-s + (−1.49 + 0.862i)3-s + (0.424 + 0.735i)4-s − 2.34·6-s + (−0.745 − 0.666i)7-s − 0.205i·8-s + (0.989 − 1.71i)9-s + (−0.316 − 0.548i)11-s + (−1.26 − 0.732i)12-s + 1.30i·13-s + (−0.424 − 1.29i)14-s + (0.564 − 0.976i)16-s + (−0.00850 + 0.00490i)17-s + (2.33 − 1.34i)18-s + (0.381 − 0.661i)19-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.315+0.949i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.315+0.949i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.315+0.949i0.315 + 0.949i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(149,⋅)\chi_{175} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), 0.315+0.949i)(2,\ 175,\ (\ :3/2),\ 0.315 + 0.949i)

Particular Values

L(2)L(2) ≈\approx 0.493707−0.356276i0.493707 - 0.356276i
L(12)L(\frac12) ≈\approx 0.493707−0.356276i0.493707 - 0.356276i
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+(13.8+12.3i)T 1 + (13.8 + 12.3i)T
good2 1+(−3.33−1.92i)T+(4+6.92i)T2 1 + (-3.33 - 1.92i)T + (4 + 6.92i)T^{2}
3 1+(7.76−4.48i)T+(13.5−23.3i)T2 1 + (7.76 - 4.48i)T + (13.5 - 23.3i)T^{2}
11 1+(11.5+20.0i)T+(−665.5+1.15e3i)T2 1 + (11.5 + 20.0i)T + (-665.5 + 1.15e3i)T^{2}
13 1−61.0iT−2.19e3T2 1 - 61.0iT - 2.19e3T^{2}
17 1+(0.596−0.344i)T+(2.45e3−4.25e3i)T2 1 + (0.596 - 0.344i)T + (2.45e3 - 4.25e3i)T^{2}
19 1+(−31.6+54.7i)T+(−3.42e3−5.94e3i)T2 1 + (-31.6 + 54.7i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(107.+62.2i)T+(6.08e3+1.05e4i)T2 1 + (107. + 62.2i)T + (6.08e3 + 1.05e4i)T^{2}
29 1+104.T+2.43e4T2 1 + 104.T + 2.43e4T^{2}
31 1+(140.+242.i)T+(−1.48e4+2.57e4i)T2 1 + (140. + 242. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(228.+131.i)T+(2.53e4+4.38e4i)T2 1 + (228. + 131. i)T + (2.53e4 + 4.38e4i)T^{2}
41 1+243.T+6.89e4T2 1 + 243.T + 6.89e4T^{2}
43 1−172.iT−7.95e4T2 1 - 172. iT - 7.95e4T^{2}
47 1+(92.9+53.6i)T+(5.19e4+8.99e4i)T2 1 + (92.9 + 53.6i)T + (5.19e4 + 8.99e4i)T^{2}
53 1+(−38.8+22.4i)T+(7.44e4−1.28e5i)T2 1 + (-38.8 + 22.4i)T + (7.44e4 - 1.28e5i)T^{2}
59 1+(−228.−395.i)T+(−1.02e5+1.77e5i)T2 1 + (-228. - 395. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(236.−410.i)T+(−1.13e5−1.96e5i)T2 1 + (236. - 410. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−198.+114.i)T+(1.50e5−2.60e5i)T2 1 + (-198. + 114. i)T + (1.50e5 - 2.60e5i)T^{2}
71 1−407.T+3.57e5T2 1 - 407.T + 3.57e5T^{2}
73 1+(301.−174.i)T+(1.94e5−3.36e5i)T2 1 + (301. - 174. i)T + (1.94e5 - 3.36e5i)T^{2}
79 1+(−420.+727.i)T+(−2.46e5−4.26e5i)T2 1 + (-420. + 727. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+885.iT−5.71e5T2 1 + 885. iT - 5.71e5T^{2}
89 1+(428.−741.i)T+(−3.52e5−6.10e5i)T2 1 + (428. - 741. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+189.iT−9.12e5T2 1 + 189. iT - 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

−12.08506232212818506846552784201, −11.25479256776260257272099569153, −10.22699147985012804248171157464, −9.353804597450380970562297371379, −7.19367404491147584129357330361, −6.35378499624710292857260192267, −5.57777706946123597598108465617, −4.47381934898725499859787078388, −3.72049546492953291748846848905, −0.21964600931817025925112850350, 1.82260437262016184238413900765, 3.41082310202111241785301488519, 5.24640452484196390400340848663, 5.58795005167364177740673920216, 6.77112437841602536748483923984, 8.073949786389934926122310282635, 10.02822005102080991770776611143, 10.87735937397874704627128389662, 11.93951237181382044145814718385, 12.42713970745056358240839113296

Graph of the ZZ-function along the critical line