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Properties

Label 2-175-175.108-c1-0-6
Degree 22
Conductor 175175
Sign 0.788−0.615i0.788 - 0.615i
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  + (0.0712 + 1.35i)2-s + (−0.508 − 0.412i)3-s + (0.146 − 0.0153i)4-s + (2.05 − 0.886i)5-s + (0.524 − 0.721i)6-s + (1.38 − 2.25i)7-s + (0.457 + 2.88i)8-s + (−0.534 − 2.51i)9-s + (1.35 + 2.72i)10-s + (−3.52 − 0.749i)11-s + (−0.0806 − 0.0523i)12-s + (3.11 + 6.10i)13-s + (3.16 + 1.72i)14-s + (−1.41 − 0.395i)15-s + (−3.60 + 0.766i)16-s + (−0.931 − 0.357i)17-s + ⋯
L(s)  = 1  + (0.0503 + 0.961i)2-s + (−0.293 − 0.237i)3-s + (0.0730 − 0.00767i)4-s + (0.918 − 0.396i)5-s + (0.213 − 0.294i)6-s + (0.523 − 0.851i)7-s + (0.161 + 1.02i)8-s + (−0.178 − 0.838i)9-s + (0.427 + 0.862i)10-s + (−1.06 − 0.225i)11-s + (−0.0232 − 0.0151i)12-s + (0.862 + 1.69i)13-s + (0.845 + 0.460i)14-s + (−0.364 − 0.101i)15-s + (−0.901 + 0.191i)16-s + (−0.225 − 0.0866i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.788−0.615i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.788−0.615i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.788−0.615i0.788 - 0.615i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(108,⋅)\chi_{175} (108, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), 0.788−0.615i)(2,\ 175,\ (\ :1/2),\ 0.788 - 0.615i)

Particular Values

L(1)L(1) ≈\approx 1.28857+0.443301i1.28857 + 0.443301i
L(12)L(\frac12) ≈\approx 1.28857+0.443301i1.28857 + 0.443301i
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+(−2.05+0.886i)T 1 + (-2.05 + 0.886i)T
7 1+(−1.38+2.25i)T 1 + (-1.38 + 2.25i)T
good2 1+(−0.0712−1.35i)T+(−1.98+0.209i)T2 1 + (-0.0712 - 1.35i)T + (-1.98 + 0.209i)T^{2}
3 1+(0.508+0.412i)T+(0.623+2.93i)T2 1 + (0.508 + 0.412i)T + (0.623 + 2.93i)T^{2}
11 1+(3.52+0.749i)T+(10.0+4.47i)T2 1 + (3.52 + 0.749i)T + (10.0 + 4.47i)T^{2}
13 1+(−3.11−6.10i)T+(−7.64+10.5i)T2 1 + (-3.11 - 6.10i)T + (-7.64 + 10.5i)T^{2}
17 1+(0.931+0.357i)T+(12.6+11.3i)T2 1 + (0.931 + 0.357i)T + (12.6 + 11.3i)T^{2}
19 1+(0.467−4.44i)T+(−18.5−3.95i)T2 1 + (0.467 - 4.44i)T + (-18.5 - 3.95i)T^{2}
23 1+(3.95−0.207i)T+(22.8−2.40i)T2 1 + (3.95 - 0.207i)T + (22.8 - 2.40i)T^{2}
29 1+(4.04+5.57i)T+(−8.96+27.5i)T2 1 + (4.04 + 5.57i)T + (-8.96 + 27.5i)T^{2}
31 1+(2.51+5.65i)T+(−20.7+23.0i)T2 1 + (2.51 + 5.65i)T + (-20.7 + 23.0i)T^{2}
37 1+(1.65−2.54i)T+(−15.0−33.8i)T2 1 + (1.65 - 2.54i)T + (-15.0 - 33.8i)T^{2}
41 1+(−5.72−1.85i)T+(33.1+24.0i)T2 1 + (-5.72 - 1.85i)T + (33.1 + 24.0i)T^{2}
43 1+(−3.08−3.08i)T+43iT2 1 + (-3.08 - 3.08i)T + 43iT^{2}
47 1+(1.75+4.56i)T+(−34.9+31.4i)T2 1 + (1.75 + 4.56i)T + (-34.9 + 31.4i)T^{2}
53 1+(5.55−6.86i)T+(−11.0−51.8i)T2 1 + (5.55 - 6.86i)T + (-11.0 - 51.8i)T^{2}
59 1+(−1.85−2.05i)T+(−6.16+58.6i)T2 1 + (-1.85 - 2.05i)T + (-6.16 + 58.6i)T^{2}
61 1+(−2.72−2.44i)T+(6.37+60.6i)T2 1 + (-2.72 - 2.44i)T + (6.37 + 60.6i)T^{2}
67 1+(−5.05+13.1i)T+(−49.7−44.8i)T2 1 + (-5.05 + 13.1i)T + (-49.7 - 44.8i)T^{2}
71 1+(7.11−5.17i)T+(21.9−67.5i)T2 1 + (7.11 - 5.17i)T + (21.9 - 67.5i)T^{2}
73 1+(4.33−2.81i)T+(29.6−66.6i)T2 1 + (4.33 - 2.81i)T + (29.6 - 66.6i)T^{2}
79 1+(0.406−0.913i)T+(−52.8−58.7i)T2 1 + (0.406 - 0.913i)T + (-52.8 - 58.7i)T^{2}
83 1+(13.6−2.16i)T+(78.9−25.6i)T2 1 + (13.6 - 2.16i)T + (78.9 - 25.6i)T^{2}
89 1+(−1.15+1.28i)T+(−9.30−88.5i)T2 1 + (-1.15 + 1.28i)T + (-9.30 - 88.5i)T^{2}
97 1+(−13.1−2.08i)T+(92.2+29.9i)T2 1 + (-13.1 - 2.08i)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

−13.12713524771302154679403616916, −11.73141298704437608708681498806, −10.95740246943546478786311227035, −9.700220265356557506633636486063, −8.516686086344693733656446627746, −7.49199616659181103536909014486, −6.32294205021792669600463757380, −5.75722185178302110862197025548, −4.27640866405497567371138621295, −1.84473769368120531680832502530, 2.05432938172655929680153286928, 3.02490650807016513382747001792, 5.10082766750482691346733564092, 5.89085995014606681182908791360, 7.48790423841381288142423600221, 8.751908641271241480440062280419, 10.12723825764002943452433950648, 10.75547845314360968957103987044, 11.24519581150436788806725788347, 12.76470835416798495731675467345

Graph of the ZZ-function along the critical line