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Properties

Label 2-175-175.103-c3-0-22
Degree 22
Conductor 175175
Sign 0.755+0.654i0.755 + 0.654i
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  + (−4.61 − 0.241i)2-s + (1.85 + 2.29i)3-s + (13.3 + 1.39i)4-s + (−10.3 + 4.28i)5-s + (−8.00 − 11.0i)6-s + (−16.5 + 8.39i)7-s + (−24.5 − 3.88i)8-s + (3.80 − 17.9i)9-s + (48.7 − 17.2i)10-s + (12.7 − 2.71i)11-s + (21.4 + 33.0i)12-s + (−54.2 − 27.6i)13-s + (78.2 − 34.7i)14-s + (−28.9 − 15.7i)15-s + (7.70 + 1.63i)16-s + (27.3 + 71.3i)17-s + ⋯
L(s)  = 1  + (−1.63 − 0.0855i)2-s + (0.356 + 0.440i)3-s + (1.66 + 0.174i)4-s + (−0.923 + 0.382i)5-s + (−0.544 − 0.749i)6-s + (−0.891 + 0.453i)7-s + (−1.08 − 0.171i)8-s + (0.141 − 0.663i)9-s + (1.54 − 0.545i)10-s + (0.350 − 0.0744i)11-s + (0.516 + 0.795i)12-s + (−1.15 − 0.589i)13-s + (1.49 − 0.663i)14-s + (−0.498 − 0.270i)15-s + (0.120 + 0.0255i)16-s + (0.390 + 1.01i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.755+0.654i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.755+0.654i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.755+0.654i0.755 + 0.654i
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), 0.755+0.654i)(2,\ 175,\ (\ :3/2),\ 0.755 + 0.654i)

Particular Values

L(2)L(2) ≈\approx 0.424621−0.158335i0.424621 - 0.158335i
L(12)L(\frac12) ≈\approx 0.424621−0.158335i0.424621 - 0.158335i
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+(10.3−4.28i)T 1 + (10.3 - 4.28i)T
7 1+(16.5−8.39i)T 1 + (16.5 - 8.39i)T
good2 1+(4.61+0.241i)T+(7.95+0.836i)T2 1 + (4.61 + 0.241i)T + (7.95 + 0.836i)T^{2}
3 1+(−1.85−2.29i)T+(−5.61+26.4i)T2 1 + (-1.85 - 2.29i)T + (-5.61 + 26.4i)T^{2}
11 1+(−12.7+2.71i)T+(1.21e3−541.i)T2 1 + (-12.7 + 2.71i)T + (1.21e3 - 541. i)T^{2}
13 1+(54.2+27.6i)T+(1.29e3+1.77e3i)T2 1 + (54.2 + 27.6i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−27.3−71.3i)T+(−3.65e3+3.28e3i)T2 1 + (-27.3 - 71.3i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−3.44−32.7i)T+(−6.70e3+1.42e3i)T2 1 + (-3.44 - 32.7i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−0.176+3.36i)T+(−1.21e4−1.27e3i)T2 1 + (-0.176 + 3.36i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(39.3−54.2i)T+(−7.53e3−2.31e4i)T2 1 + (39.3 - 54.2i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−111.+249.i)T+(−1.99e4−2.21e4i)T2 1 + (-111. + 249. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−200.+130.i)T+(2.06e4−4.62e4i)T2 1 + (-200. + 130. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−354.+115.i)T+(5.57e4−4.05e4i)T2 1 + (-354. + 115. i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(251.+251.i)T+7.95e4iT2 1 + (251. + 251. i)T + 7.95e4iT^{2}
47 1+(−402.−154.i)T+(7.71e4+6.94e4i)T2 1 + (-402. - 154. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(455.−368.i)T+(3.09e4−1.45e5i)T2 1 + (455. - 368. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−302.+335.i)T+(−2.14e4−2.04e5i)T2 1 + (-302. + 335. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−320.+288.i)T+(2.37e4−2.25e5i)T2 1 + (-320. + 288. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−43.9+16.8i)T+(2.23e5−2.01e5i)T2 1 + (-43.9 + 16.8i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(87.7+63.7i)T+(1.10e5+3.40e5i)T2 1 + (87.7 + 63.7i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−4.65+7.16i)T+(−1.58e5−3.55e5i)T2 1 + (-4.65 + 7.16i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(80.1+179.i)T+(−3.29e5+3.66e5i)T2 1 + (80.1 + 179. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−76.1+480.i)T+(−5.43e5−1.76e5i)T2 1 + (-76.1 + 480. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(103.+114.i)T+(−7.36e4+7.01e5i)T2 1 + (103. + 114. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(−48.3−305.i)T+(−8.68e5+2.82e5i)T2 1 + (-48.3 - 305. 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

−11.94645341654321201538198934721, −10.75215326685813890894056169859, −9.881047324990058401923495424221, −9.256569556098918036290548302696, −8.202621540907179782891352317703, −7.34464313655892987259230590860, −6.20099044474994205665195237330, −3.93536884489887145786368715548, −2.67451455455544107504545507196, −0.44561815509314583255583088748, 0.958477582637067680815211833952, 2.71391087817698486718601806270, 4.62174280098992441348600087143, 6.87681848174228235728126944337, 7.35597727447212039998661547635, 8.255730082440652051897233685704, 9.310353231041596471635318673382, 10.01054790913820241582654772024, 11.20822609134682940403327360747, 12.11082728193748672727585367409

Graph of the ZZ-function along the critical line