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Properties

Label 2-175-175.103-c3-0-15
Degree 22
Conductor 175175
Sign −0.970−0.241i-0.970 - 0.241i
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  + (−2.60 − 0.136i)2-s + (4.66 + 5.76i)3-s + (−1.21 − 0.127i)4-s + (1.25 + 11.1i)5-s + (−11.3 − 15.6i)6-s + (−1.27 + 18.4i)7-s + (23.7 + 3.75i)8-s + (−5.81 + 27.3i)9-s + (−1.73 − 29.0i)10-s + (−17.0 + 3.61i)11-s + (−4.91 − 7.57i)12-s + (19.8 + 10.0i)13-s + (5.83 − 47.8i)14-s + (−58.1 + 59.0i)15-s + (−51.6 − 10.9i)16-s + (40.0 + 104. i)17-s + ⋯
L(s)  = 1  + (−0.919 − 0.0481i)2-s + (0.897 + 1.10i)3-s + (−0.151 − 0.0159i)4-s + (0.111 + 0.993i)5-s + (−0.772 − 1.06i)6-s + (−0.0687 + 0.997i)7-s + (1.04 + 0.165i)8-s + (−0.215 + 1.01i)9-s + (−0.0549 − 0.919i)10-s + (−0.466 + 0.0991i)11-s + (−0.118 − 0.182i)12-s + (0.422 + 0.215i)13-s + (0.111 − 0.913i)14-s + (−1.00 + 1.01i)15-s + (−0.806 − 0.171i)16-s + (0.571 + 1.48i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.970−0.241i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.970−0.241i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.970−0.241i-0.970 - 0.241i
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), −0.970−0.241i)(2,\ 175,\ (\ :3/2),\ -0.970 - 0.241i)

Particular Values

L(2)L(2) ≈\approx 0.131284+1.07223i0.131284 + 1.07223i
L(12)L(\frac12) ≈\approx 0.131284+1.07223i0.131284 + 1.07223i
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+(−1.25−11.1i)T 1 + (-1.25 - 11.1i)T
7 1+(1.27−18.4i)T 1 + (1.27 - 18.4i)T
good2 1+(2.60+0.136i)T+(7.95+0.836i)T2 1 + (2.60 + 0.136i)T + (7.95 + 0.836i)T^{2}
3 1+(−4.66−5.76i)T+(−5.61+26.4i)T2 1 + (-4.66 - 5.76i)T + (-5.61 + 26.4i)T^{2}
11 1+(17.0−3.61i)T+(1.21e3−541.i)T2 1 + (17.0 - 3.61i)T + (1.21e3 - 541. i)T^{2}
13 1+(−19.8−10.0i)T+(1.29e3+1.77e3i)T2 1 + (-19.8 - 10.0i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−40.0−104.i)T+(−3.65e3+3.28e3i)T2 1 + (-40.0 - 104. i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(10.2+97.8i)T+(−6.70e3+1.42e3i)T2 1 + (10.2 + 97.8i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−8.15+155.i)T+(−1.21e4−1.27e3i)T2 1 + (-8.15 + 155. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(6.80−9.36i)T+(−7.53e3−2.31e4i)T2 1 + (6.80 - 9.36i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(10.0−22.5i)T+(−1.99e4−2.21e4i)T2 1 + (10.0 - 22.5i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(271.−176.i)T+(2.06e4−4.62e4i)T2 1 + (271. - 176. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(277.−90.1i)T+(5.57e4−4.05e4i)T2 1 + (277. - 90.1i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−245.−245.i)T+7.95e4iT2 1 + (-245. - 245. i)T + 7.95e4iT^{2}
47 1+(181.+69.7i)T+(7.71e4+6.94e4i)T2 1 + (181. + 69.7i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(−148.+120.i)T+(3.09e4−1.45e5i)T2 1 + (-148. + 120. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−385.+428.i)T+(−2.14e4−2.04e5i)T2 1 + (-385. + 428. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−317.+285.i)T+(2.37e4−2.25e5i)T2 1 + (-317. + 285. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(84.7−32.5i)T+(2.23e5−2.01e5i)T2 1 + (84.7 - 32.5i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−518.−376.i)T+(1.10e5+3.40e5i)T2 1 + (-518. - 376. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−137.+212.i)T+(−1.58e5−3.55e5i)T2 1 + (-137. + 212. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(−409.−920.i)T+(−3.29e5+3.66e5i)T2 1 + (-409. - 920. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−35.9+226.i)T+(−5.43e5−1.76e5i)T2 1 + (-35.9 + 226. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−378.−420.i)T+(−7.36e4+7.01e5i)T2 1 + (-378. - 420. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(20.7+130.i)T+(−8.68e5+2.82e5i)T2 1 + (20.7 + 130. i)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

−12.79507533406452623107332034443, −11.15920009167622310440690752518, −10.34413590941867189910880510356, −9.727895788505246124094809420169, −8.677850468359621804240996070301, −8.215778616989790604675524930098, −6.57627248022419153085042804013, −4.95171805430968209035831194931, −3.56093706251182781459173529685, −2.28786062489259312303652092415, 0.63146374273455866376030509221, 1.65094424928450272327293818374, 3.68684477429543834095729117957, 5.30617485583229123275079388486, 7.24320216747282874719026936907, 7.75356715565368187793500974625, 8.597324243891048859550689084446, 9.491129283483341147616977890575, 10.43782008178706209906830241022, 11.99144495276171941143667554200

Graph of the ZZ-function along the critical line