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Properties

Label 2-175-175.17-c1-0-11
Degree 22
Conductor 175175
Sign 0.981−0.193i0.981 - 0.193i
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  + (2.50 − 0.131i)2-s + (−0.971 + 1.20i)3-s + (4.26 − 0.448i)4-s + (−1.53 − 1.62i)5-s + (−2.27 + 3.13i)6-s + (2.61 + 0.377i)7-s + (5.68 − 0.899i)8-s + (0.127 + 0.601i)9-s + (−4.05 − 3.87i)10-s + (−5.16 − 1.09i)11-s + (−3.61 + 5.56i)12-s + (−2.31 + 1.17i)13-s + (6.60 + 0.602i)14-s + (3.44 − 0.255i)15-s + (5.71 − 1.21i)16-s + (0.412 − 1.07i)17-s + ⋯
L(s)  = 1  + (1.77 − 0.0928i)2-s + (−0.561 + 0.692i)3-s + (2.13 − 0.224i)4-s + (−0.685 − 0.728i)5-s + (−0.929 + 1.27i)6-s + (0.989 + 0.142i)7-s + (2.00 − 0.318i)8-s + (0.0426 + 0.200i)9-s + (−1.28 − 1.22i)10-s + (−1.55 − 0.331i)11-s + (−1.04 + 1.60i)12-s + (−0.641 + 0.326i)13-s + (1.76 + 0.161i)14-s + (0.889 − 0.0660i)15-s + (1.42 − 0.303i)16-s + (0.100 − 0.260i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.981−0.193i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.981−0.193i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.981−0.193i0.981 - 0.193i
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), 0.981−0.193i)(2,\ 175,\ (\ :1/2),\ 0.981 - 0.193i)

Particular Values

L(1)L(1) ≈\approx 2.30967+0.225414i2.30967 + 0.225414i
L(12)L(\frac12) ≈\approx 2.30967+0.225414i2.30967 + 0.225414i
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+(1.53+1.62i)T 1 + (1.53 + 1.62i)T
7 1+(−2.61−0.377i)T 1 + (-2.61 - 0.377i)T
good2 1+(−2.50+0.131i)T+(1.98−0.209i)T2 1 + (-2.50 + 0.131i)T + (1.98 - 0.209i)T^{2}
3 1+(0.971−1.20i)T+(−0.623−2.93i)T2 1 + (0.971 - 1.20i)T + (-0.623 - 2.93i)T^{2}
11 1+(5.16+1.09i)T+(10.0+4.47i)T2 1 + (5.16 + 1.09i)T + (10.0 + 4.47i)T^{2}
13 1+(2.31−1.17i)T+(7.64−10.5i)T2 1 + (2.31 - 1.17i)T + (7.64 - 10.5i)T^{2}
17 1+(−0.412+1.07i)T+(−12.6−11.3i)T2 1 + (-0.412 + 1.07i)T + (-12.6 - 11.3i)T^{2}
19 1+(−0.136+1.29i)T+(−18.5−3.95i)T2 1 + (-0.136 + 1.29i)T + (-18.5 - 3.95i)T^{2}
23 1+(0.453+8.64i)T+(−22.8+2.40i)T2 1 + (0.453 + 8.64i)T + (-22.8 + 2.40i)T^{2}
29 1+(−1.59−2.18i)T+(−8.96+27.5i)T2 1 + (-1.59 - 2.18i)T + (-8.96 + 27.5i)T^{2}
31 1+(−2.42−5.44i)T+(−20.7+23.0i)T2 1 + (-2.42 - 5.44i)T + (-20.7 + 23.0i)T^{2}
37 1+(−2.70−1.75i)T+(15.0+33.8i)T2 1 + (-2.70 - 1.75i)T + (15.0 + 33.8i)T^{2}
41 1+(−6.10−1.98i)T+(33.1+24.0i)T2 1 + (-6.10 - 1.98i)T + (33.1 + 24.0i)T^{2}
43 1+(3.37−3.37i)T−43iT2 1 + (3.37 - 3.37i)T - 43iT^{2}
47 1+(−6.65+2.55i)T+(34.9−31.4i)T2 1 + (-6.65 + 2.55i)T + (34.9 - 31.4i)T^{2}
53 1+(−2.65−2.14i)T+(11.0+51.8i)T2 1 + (-2.65 - 2.14i)T + (11.0 + 51.8i)T^{2}
59 1+(−1.17−1.30i)T+(−6.16+58.6i)T2 1 + (-1.17 - 1.30i)T + (-6.16 + 58.6i)T^{2}
61 1+(−3.10−2.79i)T+(6.37+60.6i)T2 1 + (-3.10 - 2.79i)T + (6.37 + 60.6i)T^{2}
67 1+(−0.225−0.0865i)T+(49.7+44.8i)T2 1 + (-0.225 - 0.0865i)T + (49.7 + 44.8i)T^{2}
71 1+(8.50−6.18i)T+(21.9−67.5i)T2 1 + (8.50 - 6.18i)T + (21.9 - 67.5i)T^{2}
73 1+(6.73+10.3i)T+(−29.6+66.6i)T2 1 + (6.73 + 10.3i)T + (-29.6 + 66.6i)T^{2}
79 1+(−1.88+4.22i)T+(−52.8−58.7i)T2 1 + (-1.88 + 4.22i)T + (-52.8 - 58.7i)T^{2}
83 1+(−1.46−9.26i)T+(−78.9+25.6i)T2 1 + (-1.46 - 9.26i)T + (-78.9 + 25.6i)T^{2}
89 1+(3.01−3.34i)T+(−9.30−88.5i)T2 1 + (3.01 - 3.34i)T + (-9.30 - 88.5i)T^{2}
97 1+(−1.20+7.60i)T+(−92.2−29.9i)T2 1 + (-1.20 + 7.60i)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

−12.68745073770488393843832508508, −11.93110545403336653686137749448, −11.08790335557612405714544427875, −10.40267274277463509447426060508, −8.436794413183827709883507595040, −7.34674739939828097504420818577, −5.65031349842384530072616844800, −4.79371808459749967302307987027, −4.49179630497031470603577863684, −2.63058344461571979132601863210, 2.41844300372131987112431682124, 3.88954559586875999386812792938, 5.14859146256801398995891472909, 6.02544092669229570872315258980, 7.48091960273327422528191302205, 7.61524687903706087305385458405, 10.29579687191245507324500651926, 11.33652308223909610889671618798, 11.84766348436997026926459963158, 12.72725636211556561142733255813

Graph of the ZZ-function along the critical line