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Properties

Label 2-175-175.17-c1-0-1
Degree 22
Conductor 175175
Sign −0.998−0.0463i-0.998 - 0.0463i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.857 − 0.0449i)2-s + (−1.95 + 2.41i)3-s + (−1.25 + 0.131i)4-s + (−0.521 − 2.17i)5-s + (−1.56 + 2.15i)6-s + (−2.09 + 1.61i)7-s + (−2.76 + 0.438i)8-s + (−1.38 − 6.50i)9-s + (−0.544 − 1.84i)10-s + (2.41 + 0.513i)11-s + (2.13 − 3.29i)12-s + (−3.07 + 1.56i)13-s + (−1.72 + 1.47i)14-s + (6.26 + 2.99i)15-s + (0.117 − 0.0249i)16-s + (−1.30 + 3.40i)17-s + ⋯
L(s)  = 1  + (0.606 − 0.0317i)2-s + (−1.12 + 1.39i)3-s + (−0.627 + 0.0659i)4-s + (−0.233 − 0.972i)5-s + (−0.640 + 0.880i)6-s + (−0.793 + 0.608i)7-s + (−0.978 + 0.154i)8-s + (−0.460 − 2.16i)9-s + (−0.172 − 0.582i)10-s + (0.728 + 0.154i)11-s + (0.616 − 0.949i)12-s + (−0.854 + 0.435i)13-s + (−0.461 + 0.394i)14-s + (1.61 + 0.772i)15-s + (0.0293 − 0.00624i)16-s + (−0.316 + 0.825i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.998−0.0463i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((−0.998−0.0463i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.998−0.0463i-0.998 - 0.0463i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(17,⋅)\chi_{175} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), −0.998−0.0463i)(2,\ 175,\ (\ :1/2),\ -0.998 - 0.0463i)

Particular Values

L(1)L(1) ≈\approx 0.00835839+0.360694i0.00835839 + 0.360694i
L(12)L(\frac12) ≈\approx 0.00835839+0.360694i0.00835839 + 0.360694i
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+(0.521+2.17i)T 1 + (0.521 + 2.17i)T
7 1+(2.09−1.61i)T 1 + (2.09 - 1.61i)T
good2 1+(−0.857+0.0449i)T+(1.98−0.209i)T2 1 + (-0.857 + 0.0449i)T + (1.98 - 0.209i)T^{2}
3 1+(1.95−2.41i)T+(−0.623−2.93i)T2 1 + (1.95 - 2.41i)T + (-0.623 - 2.93i)T^{2}
11 1+(−2.41−0.513i)T+(10.0+4.47i)T2 1 + (-2.41 - 0.513i)T + (10.0 + 4.47i)T^{2}
13 1+(3.07−1.56i)T+(7.64−10.5i)T2 1 + (3.07 - 1.56i)T + (7.64 - 10.5i)T^{2}
17 1+(1.30−3.40i)T+(−12.6−11.3i)T2 1 + (1.30 - 3.40i)T + (-12.6 - 11.3i)T^{2}
19 1+(0.742−7.06i)T+(−18.5−3.95i)T2 1 + (0.742 - 7.06i)T + (-18.5 - 3.95i)T^{2}
23 1+(0.107+2.04i)T+(−22.8+2.40i)T2 1 + (0.107 + 2.04i)T + (-22.8 + 2.40i)T^{2}
29 1+(−0.594−0.817i)T+(−8.96+27.5i)T2 1 + (-0.594 - 0.817i)T + (-8.96 + 27.5i)T^{2}
31 1+(1.48+3.32i)T+(−20.7+23.0i)T2 1 + (1.48 + 3.32i)T + (-20.7 + 23.0i)T^{2}
37 1+(−0.757−0.492i)T+(15.0+33.8i)T2 1 + (-0.757 - 0.492i)T + (15.0 + 33.8i)T^{2}
41 1+(−2.23−0.727i)T+(33.1+24.0i)T2 1 + (-2.23 - 0.727i)T + (33.1 + 24.0i)T^{2}
43 1+(−1.54+1.54i)T−43iT2 1 + (-1.54 + 1.54i)T - 43iT^{2}
47 1+(2.51−0.965i)T+(34.9−31.4i)T2 1 + (2.51 - 0.965i)T + (34.9 - 31.4i)T^{2}
53 1+(0.683+0.553i)T+(11.0+51.8i)T2 1 + (0.683 + 0.553i)T + (11.0 + 51.8i)T^{2}
59 1+(0.825+0.917i)T+(−6.16+58.6i)T2 1 + (0.825 + 0.917i)T + (-6.16 + 58.6i)T^{2}
61 1+(8.07+7.26i)T+(6.37+60.6i)T2 1 + (8.07 + 7.26i)T + (6.37 + 60.6i)T^{2}
67 1+(−4.72−1.81i)T+(49.7+44.8i)T2 1 + (-4.72 - 1.81i)T + (49.7 + 44.8i)T^{2}
71 1+(11.9−8.65i)T+(21.9−67.5i)T2 1 + (11.9 - 8.65i)T + (21.9 - 67.5i)T^{2}
73 1+(−4.49−6.92i)T+(−29.6+66.6i)T2 1 + (-4.49 - 6.92i)T + (-29.6 + 66.6i)T^{2}
79 1+(6.17−13.8i)T+(−52.8−58.7i)T2 1 + (6.17 - 13.8i)T + (-52.8 - 58.7i)T^{2}
83 1+(−0.481−3.03i)T+(−78.9+25.6i)T2 1 + (-0.481 - 3.03i)T + (-78.9 + 25.6i)T^{2}
89 1+(6.29−6.98i)T+(−9.30−88.5i)T2 1 + (6.29 - 6.98i)T + (-9.30 - 88.5i)T^{2}
97 1+(−2.78+17.5i)T+(−92.2−29.9i)T2 1 + (-2.78 + 17.5i)T + (-92.2 - 29.9i)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

−12.69172073578447049971300574925, −12.34874419398906326272886106015, −11.50793088314850862475931836297, −9.989060284236183874299541821012, −9.453094190283840455062574287507, −8.552885834456207763389615512429, −6.22746944501391492080689792432, −5.48912945944509543858765066423, −4.42364687106305354331073520052, −3.77899683311456966752959653427, 0.31025243709242205541084798253, 2.95102831098113879437635654741, 4.68346987071204700833172560918, 5.99968479950292101772761593265, 6.82330341598654033614592597071, 7.48493923686433028456727083717, 9.277927629555489449585438403458, 10.57048011215093599582952427779, 11.56447813369358873983280054848, 12.30176629823803375612245131208

Graph of the ZZ-function along the critical line