↑

Properties

Label 2-175-175.103-c3-0-26
Degree 22
Conductor 175175
Sign 0.687−0.725i0.687 - 0.725i
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  + (−1.21 − 0.0635i)2-s + (3.22 + 3.97i)3-s + (−6.49 − 0.682i)4-s + (8.85 + 6.81i)5-s + (−3.65 − 5.03i)6-s + (10.7 − 15.0i)7-s + (17.4 + 2.75i)8-s + (0.160 − 0.756i)9-s + (−10.3 − 8.83i)10-s + (25.0 − 5.33i)11-s + (−18.2 − 28.0i)12-s + (2.21 + 1.12i)13-s + (−13.9 + 17.6i)14-s + (1.41 + 57.2i)15-s + (30.1 + 6.40i)16-s + (6.65 + 17.3i)17-s + ⋯
L(s)  = 1  + (−0.428 − 0.0224i)2-s + (0.620 + 0.765i)3-s + (−0.811 − 0.0852i)4-s + (0.792 + 0.609i)5-s + (−0.248 − 0.342i)6-s + (0.579 − 0.814i)7-s + (0.769 + 0.121i)8-s + (0.00595 − 0.0280i)9-s + (−0.325 − 0.279i)10-s + (0.687 − 0.146i)11-s + (−0.437 − 0.674i)12-s + (0.0471 + 0.0240i)13-s + (−0.266 + 0.336i)14-s + (0.0242 + 0.985i)15-s + (0.470 + 0.100i)16-s + (0.0950 + 0.247i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.687−0.725i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.687−0.725i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.687 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.687−0.725i0.687 - 0.725i
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), 0.687−0.725i)(2,\ 175,\ (\ :3/2),\ 0.687 - 0.725i)

Particular Values

L(2)L(2) ≈\approx 1.64103+0.705573i1.64103 + 0.705573i
L(12)L(\frac12) ≈\approx 1.64103+0.705573i1.64103 + 0.705573i
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+(−8.85−6.81i)T 1 + (-8.85 - 6.81i)T
7 1+(−10.7+15.0i)T 1 + (-10.7 + 15.0i)T
good2 1+(1.21+0.0635i)T+(7.95+0.836i)T2 1 + (1.21 + 0.0635i)T + (7.95 + 0.836i)T^{2}
3 1+(−3.22−3.97i)T+(−5.61+26.4i)T2 1 + (-3.22 - 3.97i)T + (-5.61 + 26.4i)T^{2}
11 1+(−25.0+5.33i)T+(1.21e3−541.i)T2 1 + (-25.0 + 5.33i)T + (1.21e3 - 541. i)T^{2}
13 1+(−2.21−1.12i)T+(1.29e3+1.77e3i)T2 1 + (-2.21 - 1.12i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−6.65−17.3i)T+(−3.65e3+3.28e3i)T2 1 + (-6.65 - 17.3i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−13.6−130.i)T+(−6.70e3+1.42e3i)T2 1 + (-13.6 - 130. i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−1.98+37.9i)T+(−1.21e4−1.27e3i)T2 1 + (-1.98 + 37.9i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(27.5−37.9i)T+(−7.53e3−2.31e4i)T2 1 + (27.5 - 37.9i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(45.2−101.i)T+(−1.99e4−2.21e4i)T2 1 + (45.2 - 101. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(96.5−62.7i)T+(2.06e4−4.62e4i)T2 1 + (96.5 - 62.7i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−463.+150.i)T+(5.57e4−4.05e4i)T2 1 + (-463. + 150. i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−91.2−91.2i)T+7.95e4iT2 1 + (-91.2 - 91.2i)T + 7.95e4iT^{2}
47 1+(−201.−77.3i)T+(7.71e4+6.94e4i)T2 1 + (-201. - 77.3i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(232.−187.i)T+(3.09e4−1.45e5i)T2 1 + (232. - 187. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(313.−348.i)T+(−2.14e4−2.04e5i)T2 1 + (313. - 348. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−490.+441.i)T+(2.37e4−2.25e5i)T2 1 + (-490. + 441. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(142.−54.7i)T+(2.23e5−2.01e5i)T2 1 + (142. - 54.7i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(557.+405.i)T+(1.10e5+3.40e5i)T2 1 + (557. + 405. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−108.+167.i)T+(−1.58e5−3.55e5i)T2 1 + (-108. + 167. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(528.+1.18e3i)T+(−3.29e5+3.66e5i)T2 1 + (528. + 1.18e3i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(60.7−383.i)T+(−5.43e5−1.76e5i)T2 1 + (60.7 - 383. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(1.00e3+1.11e3i)T+(−7.36e4+7.01e5i)T2 1 + (1.00e3 + 1.11e3i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(−84.4−533.i)T+(−8.68e5+2.82e5i)T2 1 + (-84.4 - 533. 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

−12.48341416802050044296175256205, −10.84943205535323643748933454558, −10.21047963491905501778713655585, −9.444148337357982271833843477075, −8.596625233756386875221916585550, −7.43216373637957795082602225309, −5.93020965300431613268351961489, −4.43383125010698695010298193138, −3.50272746605721708017582947447, −1.40166099214814650338166074264, 1.13536167196453605742406021270, 2.39607049301448930895729096845, 4.53269030032997146632199740623, 5.59088717755688551333891935591, 7.19391733006182759610561493126, 8.278411450619936149278449454068, 9.012545293731224465911568573384, 9.610450185089172640797664761405, 11.14820050028182932562657175608, 12.45097514040212413090144606350

Graph of the ZZ-function along the critical line