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Properties

Label 2-175-175.103-c3-0-10
Degree 22
Conductor 175175
Sign 0.497−0.867i0.497 - 0.867i
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  + (1.99 + 0.104i)2-s + (−5.07 − 6.26i)3-s + (−3.98 − 0.419i)4-s + (−10.9 + 2.26i)5-s + (−9.46 − 13.0i)6-s + (15.9 + 9.35i)7-s + (−23.6 − 3.75i)8-s + (−7.90 + 37.1i)9-s + (−22.0 + 3.36i)10-s + (24.3 − 5.17i)11-s + (17.6 + 27.1i)12-s + (29.1 + 14.8i)13-s + (30.9 + 20.3i)14-s + (69.7 + 57.1i)15-s + (−15.4 − 3.29i)16-s + (−16.7 − 43.7i)17-s + ⋯
L(s)  = 1  + (0.705 + 0.0369i)2-s + (−0.976 − 1.20i)3-s + (−0.498 − 0.0523i)4-s + (−0.979 + 0.202i)5-s + (−0.644 − 0.886i)6-s + (0.862 + 0.505i)7-s + (−1.04 − 0.165i)8-s + (−0.292 + 1.37i)9-s + (−0.698 + 0.106i)10-s + (0.667 − 0.141i)11-s + (0.423 + 0.652i)12-s + (0.622 + 0.317i)13-s + (0.589 + 0.388i)14-s + (1.20 + 0.983i)15-s + (−0.242 − 0.0514i)16-s + (−0.239 − 0.623i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.497−0.867i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.497−0.867i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.497−0.867i0.497 - 0.867i
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), 0.497−0.867i)(2,\ 175,\ (\ :3/2),\ 0.497 - 0.867i)

Particular Values

L(2)L(2) ≈\approx 0.667798+0.386679i0.667798 + 0.386679i
L(12)L(\frac12) ≈\approx 0.667798+0.386679i0.667798 + 0.386679i
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+(10.9−2.26i)T 1 + (10.9 - 2.26i)T
7 1+(−15.9−9.35i)T 1 + (-15.9 - 9.35i)T
good2 1+(−1.99−0.104i)T+(7.95+0.836i)T2 1 + (-1.99 - 0.104i)T + (7.95 + 0.836i)T^{2}
3 1+(5.07+6.26i)T+(−5.61+26.4i)T2 1 + (5.07 + 6.26i)T + (-5.61 + 26.4i)T^{2}
11 1+(−24.3+5.17i)T+(1.21e3−541.i)T2 1 + (-24.3 + 5.17i)T + (1.21e3 - 541. i)T^{2}
13 1+(−29.1−14.8i)T+(1.29e3+1.77e3i)T2 1 + (-29.1 - 14.8i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(16.7+43.7i)T+(−3.65e3+3.28e3i)T2 1 + (16.7 + 43.7i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−3.73−35.5i)T+(−6.70e3+1.42e3i)T2 1 + (-3.73 - 35.5i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(11.4−218.i)T+(−1.21e4−1.27e3i)T2 1 + (11.4 - 218. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(130.−179.i)T+(−7.53e3−2.31e4i)T2 1 + (130. - 179. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−38.2+85.8i)T+(−1.99e4−2.21e4i)T2 1 + (-38.2 + 85.8i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(218.−141.i)T+(2.06e4−4.62e4i)T2 1 + (218. - 141. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−36.1+11.7i)T+(5.57e4−4.05e4i)T2 1 + (-36.1 + 11.7i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−119.−119.i)T+7.95e4iT2 1 + (-119. - 119. i)T + 7.95e4iT^{2}
47 1+(340.+130.i)T+(7.71e4+6.94e4i)T2 1 + (340. + 130. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(224.−181.i)T+(3.09e4−1.45e5i)T2 1 + (224. - 181. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(392.−435.i)T+(−2.14e4−2.04e5i)T2 1 + (392. - 435. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−650.+585.i)T+(2.37e4−2.25e5i)T2 1 + (-650. + 585. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(586.−225.i)T+(2.23e5−2.01e5i)T2 1 + (586. - 225. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−783.−569.i)T+(1.10e5+3.40e5i)T2 1 + (-783. - 569. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(96.8−149.i)T+(−1.58e5−3.55e5i)T2 1 + (96.8 - 149. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(128.+289.i)T+(−3.29e5+3.66e5i)T2 1 + (128. + 289. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−129.+819.i)T+(−5.43e5−1.76e5i)T2 1 + (-129. + 819. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−741.−823.i)T+(−7.36e4+7.01e5i)T2 1 + (-741. - 823. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(−35.9−226.i)T+(−8.68e5+2.82e5i)T2 1 + (-35.9 - 226. i)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

−12.27614285286908604501844255589, −11.66157934007288600125349288777, −11.17630694241959436138186052628, −9.190729167976256198074417697590, −8.070590272879412816223406255998, −7.03280531367928166600405101006, −5.91384173699896346604626647341, −4.98032359795220412107836473226, −3.60889845611392371008733093458, −1.34257482556512106426821620744, 0.36508346878130150013121490396, 3.75856272016217531664709493375, 4.34722212751535893543575531973, 5.10928050790918127037660439613, 6.40144734632431423821870812706, 8.152196483407548596378831222292, 9.071005862482601775216264889747, 10.44367345375708813088476170585, 11.18608451691648590923732454829, 11.95583115048023684385273986299

Graph of the ZZ-function along the critical line