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Properties

Label 2-175-175.103-c3-0-16
Degree 22
Conductor 175175
Sign 0.750−0.660i0.750 - 0.660i
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  + (−0.570 − 0.0299i)2-s + (1.41 + 1.74i)3-s + (−7.63 − 0.802i)4-s + (−10.8 − 2.77i)5-s + (−0.756 − 1.04i)6-s + (18.4 − 0.955i)7-s + (8.84 + 1.40i)8-s + (4.55 − 21.4i)9-s + (6.09 + 1.90i)10-s + (−42.9 + 9.13i)11-s + (−9.41 − 14.4i)12-s + (63.7 + 32.4i)13-s + (−10.5 − 0.00760i)14-s + (−10.4 − 22.8i)15-s + (55.0 + 11.6i)16-s + (44.2 + 115. i)17-s + ⋯
L(s)  = 1  + (−0.201 − 0.0105i)2-s + (0.272 + 0.336i)3-s + (−0.953 − 0.100i)4-s + (−0.968 − 0.248i)5-s + (−0.0514 − 0.0708i)6-s + (0.998 − 0.0516i)7-s + (0.390 + 0.0619i)8-s + (0.168 − 0.794i)9-s + (0.192 + 0.0603i)10-s + (−1.17 + 0.250i)11-s + (−0.226 − 0.348i)12-s + (1.35 + 0.692i)13-s + (−0.202 − 0.000145i)14-s + (−0.180 − 0.393i)15-s + (0.860 + 0.182i)16-s + (0.631 + 1.64i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.750−0.660i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.750−0.660i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.750−0.660i0.750 - 0.660i
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), 0.750−0.660i)(2,\ 175,\ (\ :3/2),\ 0.750 - 0.660i)

Particular Values

L(2)L(2) ≈\approx 1.11841+0.421948i1.11841 + 0.421948i
L(12)L(\frac12) ≈\approx 1.11841+0.421948i1.11841 + 0.421948i
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+(10.8+2.77i)T 1 + (10.8 + 2.77i)T
7 1+(−18.4+0.955i)T 1 + (-18.4 + 0.955i)T
good2 1+(0.570+0.0299i)T+(7.95+0.836i)T2 1 + (0.570 + 0.0299i)T + (7.95 + 0.836i)T^{2}
3 1+(−1.41−1.74i)T+(−5.61+26.4i)T2 1 + (-1.41 - 1.74i)T + (-5.61 + 26.4i)T^{2}
11 1+(42.9−9.13i)T+(1.21e3−541.i)T2 1 + (42.9 - 9.13i)T + (1.21e3 - 541. i)T^{2}
13 1+(−63.7−32.4i)T+(1.29e3+1.77e3i)T2 1 + (-63.7 - 32.4i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−44.2−115.i)T+(−3.65e3+3.28e3i)T2 1 + (-44.2 - 115. i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(2.03+19.3i)T+(−6.70e3+1.42e3i)T2 1 + (2.03 + 19.3i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(5.56−106.i)T+(−1.21e4−1.27e3i)T2 1 + (5.56 - 106. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(0.465−0.640i)T+(−7.53e3−2.31e4i)T2 1 + (0.465 - 0.640i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−2.69+6.05i)T+(−1.99e4−2.21e4i)T2 1 + (-2.69 + 6.05i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−305.+198.i)T+(2.06e4−4.62e4i)T2 1 + (-305. + 198. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(113.−36.7i)T+(5.57e4−4.05e4i)T2 1 + (113. - 36.7i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−18.9−18.9i)T+7.95e4iT2 1 + (-18.9 - 18.9i)T + 7.95e4iT^{2}
47 1+(−525.−201.i)T+(7.71e4+6.94e4i)T2 1 + (-525. - 201. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(109.−89.0i)T+(3.09e4−1.45e5i)T2 1 + (109. - 89.0i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−499.+554.i)T+(−2.14e4−2.04e5i)T2 1 + (-499. + 554. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(430.−387.i)T+(2.37e4−2.25e5i)T2 1 + (430. - 387. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(389.−149.i)T+(2.23e5−2.01e5i)T2 1 + (389. - 149. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−107.−78.3i)T+(1.10e5+3.40e5i)T2 1 + (-107. - 78.3i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(245.−377.i)T+(−1.58e5−3.55e5i)T2 1 + (245. - 377. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(50.0+112.i)T+(−3.29e5+3.66e5i)T2 1 + (50.0 + 112. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(23.6−149.i)T+(−5.43e5−1.76e5i)T2 1 + (23.6 - 149. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−393.−437.i)T+(−7.36e4+7.01e5i)T2 1 + (-393. - 437. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(221.+1.39e3i)T+(−8.68e5+2.82e5i)T2 1 + (221. + 1.39e3i)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

−12.44315280858663289085522143865, −11.24479204952444672362364110701, −10.38399044284884828825527566878, −9.115684148002768865350421167145, −8.360789972555113442430544432695, −7.64482271515711812814414700831, −5.73735038490609356170994001591, −4.39247305426852343218582258434, −3.72553138361596334432583966665, −1.15768507250732171109779062139, 0.76106737215995713429228777714, 2.93257870759969390599458755534, 4.45992177072927893520872812543, 5.39752399278519783236557611689, 7.48712554445071272019903792777, 8.043379824662921419657158937483, 8.665177168546088139732537860791, 10.32881498224520909980291220392, 11.01547786940007034672067362815, 12.18170316688670297223202706026

Graph of the ZZ-function along the critical line