↑

Properties

Label 2-175-175.103-c3-0-17
Degree 22
Conductor 175175
Sign 0.757+0.652i0.757 + 0.652i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.757 − 0.0397i)2-s + (−6.12 − 7.56i)3-s + (−7.38 − 0.776i)4-s + (10.8 + 2.49i)5-s + (4.34 + 5.97i)6-s + (2.54 + 18.3i)7-s + (11.5 + 1.83i)8-s + (−14.0 + 66.3i)9-s + (−8.16 − 2.32i)10-s + (32.3 − 6.87i)11-s + (39.3 + 60.6i)12-s + (−26.8 − 13.6i)13-s + (−1.20 − 14.0i)14-s + (−47.8 − 97.7i)15-s + (49.4 + 10.5i)16-s + (−20.5 − 53.5i)17-s + ⋯
L(s)  = 1  + (−0.267 − 0.0140i)2-s + (−1.17 − 1.45i)3-s + (−0.922 − 0.0970i)4-s + (0.974 + 0.223i)5-s + (0.295 + 0.406i)6-s + (0.137 + 0.990i)7-s + (0.511 + 0.0809i)8-s + (−0.522 + 2.45i)9-s + (−0.258 − 0.0735i)10-s + (0.887 − 0.188i)11-s + (0.947 + 1.45i)12-s + (−0.571 − 0.291i)13-s + (−0.0229 − 0.267i)14-s + (−0.824 − 1.68i)15-s + (0.771 + 0.164i)16-s + (−0.293 − 0.763i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.757+0.652i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.757+0.652i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.757+0.652i0.757 + 0.652i
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), 0.757+0.652i)(2,\ 175,\ (\ :3/2),\ 0.757 + 0.652i)

Particular Values

L(2)L(2) ≈\approx 0.853093−0.316848i0.853093 - 0.316848i
L(12)L(\frac12) ≈\approx 0.853093−0.316848i0.853093 - 0.316848i
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+(−10.8−2.49i)T 1 + (-10.8 - 2.49i)T
7 1+(−2.54−18.3i)T 1 + (-2.54 - 18.3i)T
good2 1+(0.757+0.0397i)T+(7.95+0.836i)T2 1 + (0.757 + 0.0397i)T + (7.95 + 0.836i)T^{2}
3 1+(6.12+7.56i)T+(−5.61+26.4i)T2 1 + (6.12 + 7.56i)T + (-5.61 + 26.4i)T^{2}
11 1+(−32.3+6.87i)T+(1.21e3−541.i)T2 1 + (-32.3 + 6.87i)T + (1.21e3 - 541. i)T^{2}
13 1+(26.8+13.6i)T+(1.29e3+1.77e3i)T2 1 + (26.8 + 13.6i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(20.5+53.5i)T+(−3.65e3+3.28e3i)T2 1 + (20.5 + 53.5i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−7.59−72.2i)T+(−6.70e3+1.42e3i)T2 1 + (-7.59 - 72.2i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−2.68+51.2i)T+(−1.21e4−1.27e3i)T2 1 + (-2.68 + 51.2i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(−132.+182.i)T+(−7.53e3−2.31e4i)T2 1 + (-132. + 182. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(109.−246.i)T+(−1.99e4−2.21e4i)T2 1 + (109. - 246. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−233.+151.i)T+(2.06e4−4.62e4i)T2 1 + (-233. + 151. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(145.−47.2i)T+(5.57e4−4.05e4i)T2 1 + (145. - 47.2i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−234.−234.i)T+7.95e4iT2 1 + (-234. - 234. i)T + 7.95e4iT^{2}
47 1+(−276.−106.i)T+(7.71e4+6.94e4i)T2 1 + (-276. - 106. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(−407.+329.i)T+(3.09e4−1.45e5i)T2 1 + (-407. + 329. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−371.+412.i)T+(−2.14e4−2.04e5i)T2 1 + (-371. + 412. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−79.3+71.4i)T+(2.37e4−2.25e5i)T2 1 + (-79.3 + 71.4i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(69.2−26.5i)T+(2.23e5−2.01e5i)T2 1 + (69.2 - 26.5i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−646.−469.i)T+(1.10e5+3.40e5i)T2 1 + (-646. - 469. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−382.+589.i)T+(−1.58e5−3.55e5i)T2 1 + (-382. + 589. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(−194.−437.i)T+(−3.29e5+3.66e5i)T2 1 + (-194. - 437. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−67.8+428.i)T+(−5.43e5−1.76e5i)T2 1 + (-67.8 + 428. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−85.3−94.8i)T+(−7.36e4+7.01e5i)T2 1 + (-85.3 - 94.8i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(35.2+222.i)T+(−8.68e5+2.82e5i)T2 1 + (35.2 + 222. i)T + (-8.68e5 + 2.82e5i)T^{2}
show more
show less
   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

−12.30502522171108415620363827891, −11.34233902167158140212201912993, −10.16082270629189016372403543101, −9.080954804826475836321661802271, −7.967405839312511822671719253554, −6.67192941310378496845196856562, −5.79960559619093912785227077806, −5.00326723070856035257704091201, −2.24142012288609153705417077844, −0.903329193567513662995772088935, 0.822072578539898919132865838148, 3.94915656489743683191416814468, 4.64108738406049609812787298815, 5.64024260347428536558073616871, 6.89910262988817934581033680730, 8.875219732330704889157530535636, 9.547826061461351232370747490359, 10.23528159415982167662239245619, 11.01457843681060086073667071727, 12.22470066308115824592911841660

Graph of the ZZ-function along the critical line