↑

Properties

Label 2-175-175.103-c1-0-8
Degree 22
Conductor 175175
Sign −0.502+0.864i-0.502 + 0.864i
Analytic cond. 1.397381.39738
Root an. cond. 1.182101.18210
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.08 − 0.109i)2-s + (−0.522 − 0.645i)3-s + (2.35 + 0.247i)4-s + (1.15 − 1.91i)5-s + (1.02 + 1.40i)6-s + (−1.53 + 2.15i)7-s + (−0.767 − 0.121i)8-s + (0.480 − 2.25i)9-s + (−2.61 + 3.87i)10-s + (1.02 − 0.218i)11-s + (−1.07 − 1.65i)12-s + (−4.08 − 2.08i)13-s + (3.43 − 4.33i)14-s + (−1.83 + 0.259i)15-s + (−3.05 − 0.648i)16-s + (−0.808 − 2.10i)17-s + ⋯
L(s)  = 1  + (−1.47 − 0.0773i)2-s + (−0.301 − 0.372i)3-s + (1.17 + 0.123i)4-s + (0.514 − 0.857i)5-s + (0.416 + 0.573i)6-s + (−0.578 + 0.815i)7-s + (−0.271 − 0.0429i)8-s + (0.160 − 0.753i)9-s + (−0.826 + 1.22i)10-s + (0.309 − 0.0658i)11-s + (−0.309 − 0.476i)12-s + (−1.13 − 0.576i)13-s + (0.917 − 1.15i)14-s + (−0.474 + 0.0669i)15-s + (−0.762 − 0.162i)16-s + (−0.196 − 0.510i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.502+0.864i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((−0.502+0.864i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.502+0.864i-0.502 + 0.864i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(103,⋅)\chi_{175} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), −0.502+0.864i)(2,\ 175,\ (\ :1/2),\ -0.502 + 0.864i)

Particular Values

L(1)L(1) ≈\approx 0.207276−0.360298i0.207276 - 0.360298i
L(12)L(\frac12) ≈\approx 0.207276−0.360298i0.207276 - 0.360298i
L(32)L(\frac{3}{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+(−1.15+1.91i)T 1 + (-1.15 + 1.91i)T
7 1+(1.53−2.15i)T 1 + (1.53 - 2.15i)T
good2 1+(2.08+0.109i)T+(1.98+0.209i)T2 1 + (2.08 + 0.109i)T + (1.98 + 0.209i)T^{2}
3 1+(0.522+0.645i)T+(−0.623+2.93i)T2 1 + (0.522 + 0.645i)T + (-0.623 + 2.93i)T^{2}
11 1+(−1.02+0.218i)T+(10.0−4.47i)T2 1 + (-1.02 + 0.218i)T + (10.0 - 4.47i)T^{2}
13 1+(4.08+2.08i)T+(7.64+10.5i)T2 1 + (4.08 + 2.08i)T + (7.64 + 10.5i)T^{2}
17 1+(0.808+2.10i)T+(−12.6+11.3i)T2 1 + (0.808 + 2.10i)T + (-12.6 + 11.3i)T^{2}
19 1+(0.520+4.95i)T+(−18.5+3.95i)T2 1 + (0.520 + 4.95i)T + (-18.5 + 3.95i)T^{2}
23 1+(−0.325+6.21i)T+(−22.8−2.40i)T2 1 + (-0.325 + 6.21i)T + (-22.8 - 2.40i)T^{2}
29 1+(0.743−1.02i)T+(−8.96−27.5i)T2 1 + (0.743 - 1.02i)T + (-8.96 - 27.5i)T^{2}
31 1+(4.14−9.30i)T+(−20.7−23.0i)T2 1 + (4.14 - 9.30i)T + (-20.7 - 23.0i)T^{2}
37 1+(−5.27+3.42i)T+(15.0−33.8i)T2 1 + (-5.27 + 3.42i)T + (15.0 - 33.8i)T^{2}
41 1+(−5.32+1.72i)T+(33.1−24.0i)T2 1 + (-5.32 + 1.72i)T + (33.1 - 24.0i)T^{2}
43 1+(−4.80−4.80i)T+43iT2 1 + (-4.80 - 4.80i)T + 43iT^{2}
47 1+(−0.292−0.112i)T+(34.9+31.4i)T2 1 + (-0.292 - 0.112i)T + (34.9 + 31.4i)T^{2}
53 1+(0.217−0.176i)T+(11.0−51.8i)T2 1 + (0.217 - 0.176i)T + (11.0 - 51.8i)T^{2}
59 1+(−4.60+5.10i)T+(−6.16−58.6i)T2 1 + (-4.60 + 5.10i)T + (-6.16 - 58.6i)T^{2}
61 1+(3.94−3.54i)T+(6.37−60.6i)T2 1 + (3.94 - 3.54i)T + (6.37 - 60.6i)T^{2}
67 1+(−11.4+4.40i)T+(49.7−44.8i)T2 1 + (-11.4 + 4.40i)T + (49.7 - 44.8i)T^{2}
71 1+(1.66+1.21i)T+(21.9+67.5i)T2 1 + (1.66 + 1.21i)T + (21.9 + 67.5i)T^{2}
73 1+(0.705−1.08i)T+(−29.6−66.6i)T2 1 + (0.705 - 1.08i)T + (-29.6 - 66.6i)T^{2}
79 1+(−4.40−9.89i)T+(−52.8+58.7i)T2 1 + (-4.40 - 9.89i)T + (-52.8 + 58.7i)T^{2}
83 1+(−0.579+3.66i)T+(−78.9−25.6i)T2 1 + (-0.579 + 3.66i)T + (-78.9 - 25.6i)T^{2}
89 1+(7.09+7.88i)T+(−9.30+88.5i)T2 1 + (7.09 + 7.88i)T + (-9.30 + 88.5i)T^{2}
97 1+(−0.775−4.89i)T+(−92.2+29.9i)T2 1 + (-0.775 - 4.89i)T + (-92.2 + 29.9i)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

−12.44726637983619033377961124412, −11.26804222401800146732788054320, −9.995899769024630352779612250690, −9.219726101741418026462953246107, −8.747384081224042864929566871157, −7.31373198820898349899991275549, −6.32865192128933979834614412303, −4.93845354345741177491605910609, −2.44336130855628570487791401299, −0.62528622972363830762816323416, 2.04077445859095896719620692889, 4.10657378085350231198272163881, 5.96133894689628738653610492216, 7.19519902023888544789334216266, 7.76952956264058566377823265653, 9.531803884530938415456409895221, 9.841424577074642953057877984580, 10.70319588278993512729038435537, 11.46345889475111038229130662719, 13.11399039740614589734375981634

Graph of the ZZ-function along the critical line