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Properties

Label 2-175-175.103-c3-0-23
Degree 22
Conductor 175175
Sign 0.978+0.204i0.978 + 0.204i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.143 + 0.00750i)2-s + (−0.396 − 0.489i)3-s + (−7.93 − 0.834i)4-s + (−6.20 + 9.30i)5-s + (−0.0531 − 0.0731i)6-s + (−18.1 − 3.69i)7-s + (−2.26 − 0.358i)8-s + (5.53 − 26.0i)9-s + (−0.958 + 1.28i)10-s + (62.9 − 13.3i)11-s + (2.73 + 4.21i)12-s + (50.0 + 25.5i)13-s + (−2.57 − 0.665i)14-s + (7.01 − 0.649i)15-s + (62.1 + 13.2i)16-s + (7.31 + 19.0i)17-s + ⋯
L(s)  = 1  + (0.0506 + 0.00265i)2-s + (−0.0763 − 0.0942i)3-s + (−0.991 − 0.104i)4-s + (−0.555 + 0.831i)5-s + (−0.00361 − 0.00497i)6-s + (−0.979 − 0.199i)7-s + (−0.100 − 0.0158i)8-s + (0.204 − 0.963i)9-s + (−0.0303 + 0.0406i)10-s + (1.72 − 0.366i)11-s + (0.0659 + 0.101i)12-s + (1.06 + 0.544i)13-s + (−0.0490 − 0.0126i)14-s + (0.120 − 0.0111i)15-s + (0.970 + 0.206i)16-s + (0.104 + 0.272i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.978+0.204i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.978+0.204i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.978+0.204i0.978 + 0.204i
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), 0.978+0.204i)(2,\ 175,\ (\ :3/2),\ 0.978 + 0.204i)

Particular Values

L(2)L(2) ≈\approx 1.16563−0.120336i1.16563 - 0.120336i
L(12)L(\frac12) ≈\approx 1.16563−0.120336i1.16563 - 0.120336i
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+(6.20−9.30i)T 1 + (6.20 - 9.30i)T
7 1+(18.1+3.69i)T 1 + (18.1 + 3.69i)T
good2 1+(−0.143−0.00750i)T+(7.95+0.836i)T2 1 + (-0.143 - 0.00750i)T + (7.95 + 0.836i)T^{2}
3 1+(0.396+0.489i)T+(−5.61+26.4i)T2 1 + (0.396 + 0.489i)T + (-5.61 + 26.4i)T^{2}
11 1+(−62.9+13.3i)T+(1.21e3−541.i)T2 1 + (-62.9 + 13.3i)T + (1.21e3 - 541. i)T^{2}
13 1+(−50.0−25.5i)T+(1.29e3+1.77e3i)T2 1 + (-50.0 - 25.5i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−7.31−19.0i)T+(−3.65e3+3.28e3i)T2 1 + (-7.31 - 19.0i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−7.32−69.7i)T+(−6.70e3+1.42e3i)T2 1 + (-7.32 - 69.7i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−2.91+55.6i)T+(−1.21e4−1.27e3i)T2 1 + (-2.91 + 55.6i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(−56.0+77.1i)T+(−7.53e3−2.31e4i)T2 1 + (-56.0 + 77.1i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−30.4+68.3i)T+(−1.99e4−2.21e4i)T2 1 + (-30.4 + 68.3i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−126.+82.1i)T+(2.06e4−4.62e4i)T2 1 + (-126. + 82.1i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(214.−69.7i)T+(5.57e4−4.05e4i)T2 1 + (214. - 69.7i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−269.−269.i)T+7.95e4iT2 1 + (-269. - 269. i)T + 7.95e4iT^{2}
47 1+(285.+109.i)T+(7.71e4+6.94e4i)T2 1 + (285. + 109. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(−326.+264.i)T+(3.09e4−1.45e5i)T2 1 + (-326. + 264. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−420.+467.i)T+(−2.14e4−2.04e5i)T2 1 + (-420. + 467. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−481.+433.i)T+(2.37e4−2.25e5i)T2 1 + (-481. + 433. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−1.63+0.625i)T+(2.23e5−2.01e5i)T2 1 + (-1.63 + 0.625i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−663.−481.i)T+(1.10e5+3.40e5i)T2 1 + (-663. - 481. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−140.+216.i)T+(−1.58e5−3.55e5i)T2 1 + (-140. + 216. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(250.+563.i)T+(−3.29e5+3.66e5i)T2 1 + (250. + 563. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(99.9−631.i)T+(−5.43e5−1.76e5i)T2 1 + (99.9 - 631. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−628.−698.i)T+(−7.36e4+7.01e5i)T2 1 + (-628. - 698. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(−170.−1.07e3i)T+(−8.68e5+2.82e5i)T2 1 + (-170. - 1.07e3i)T + (-8.68e5 + 2.82e5i)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

−12.20766128862003190414345602975, −11.37246432104077993084016054888, −10.01979915853730969964323953226, −9.293372682881540912899519786458, −8.271763594143884744991386007688, −6.59393675495045044956350745064, −6.22872861946262901394067377761, −3.87858692802877991054756450103, −3.69326057012525272178296949364, −0.826197120608440299810110480705, 0.966812023803598229441343558297, 3.53402369829251951271696353131, 4.48134931710315621304914687644, 5.64573169344758219950288774736, 7.12753372436396343388966611264, 8.521586191601331034402362795963, 9.093123101646588272281242200211, 10.07137561793641155075542824022, 11.48798809535879775305268476032, 12.43428393576851553058707797678

Graph of the ZZ-function along the critical line