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Properties

Label 2-175-175.103-c3-0-14
Degree 22
Conductor 175175
Sign 0.792−0.609i0.792 - 0.609i
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  + (−4.23 − 0.221i)2-s + (−3.57 − 4.41i)3-s + (9.93 + 1.04i)4-s + (11.1 + 0.705i)5-s + (14.1 + 19.4i)6-s + (17.1 + 7.03i)7-s + (−8.34 − 1.32i)8-s + (−1.09 + 5.14i)9-s + (−47.1 − 5.46i)10-s + (−65.1 + 13.8i)11-s + (−30.9 − 47.5i)12-s + (−2.69 − 1.37i)13-s + (−71.0 − 33.6i)14-s + (−36.7 − 51.7i)15-s + (−43.1 − 9.16i)16-s + (49.9 + 130. i)17-s + ⋯
L(s)  = 1  + (−1.49 − 0.0784i)2-s + (−0.687 − 0.849i)3-s + (1.24 + 0.130i)4-s + (0.998 + 0.0630i)5-s + (0.963 + 1.32i)6-s + (0.925 + 0.379i)7-s + (−0.368 − 0.0584i)8-s + (−0.0404 + 0.190i)9-s + (−1.48 − 0.172i)10-s + (−1.78 + 0.379i)11-s + (−0.743 − 1.14i)12-s + (−0.0575 − 0.0293i)13-s + (−1.35 − 0.641i)14-s + (−0.632 − 0.891i)15-s + (−0.673 − 0.143i)16-s + (0.712 + 1.85i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.792−0.609i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.792−0.609i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.792−0.609i0.792 - 0.609i
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), 0.792−0.609i)(2,\ 175,\ (\ :3/2),\ 0.792 - 0.609i)

Particular Values

L(2)L(2) ≈\approx 0.573966+0.195261i0.573966 + 0.195261i
L(12)L(\frac12) ≈\approx 0.573966+0.195261i0.573966 + 0.195261i
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+(−11.1−0.705i)T 1 + (-11.1 - 0.705i)T
7 1+(−17.1−7.03i)T 1 + (-17.1 - 7.03i)T
good2 1+(4.23+0.221i)T+(7.95+0.836i)T2 1 + (4.23 + 0.221i)T + (7.95 + 0.836i)T^{2}
3 1+(3.57+4.41i)T+(−5.61+26.4i)T2 1 + (3.57 + 4.41i)T + (-5.61 + 26.4i)T^{2}
11 1+(65.1−13.8i)T+(1.21e3−541.i)T2 1 + (65.1 - 13.8i)T + (1.21e3 - 541. i)T^{2}
13 1+(2.69+1.37i)T+(1.29e3+1.77e3i)T2 1 + (2.69 + 1.37i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−49.9−130.i)T+(−3.65e3+3.28e3i)T2 1 + (-49.9 - 130. i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−3.14−29.9i)T+(−6.70e3+1.42e3i)T2 1 + (-3.14 - 29.9i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−3.34+63.8i)T+(−1.21e4−1.27e3i)T2 1 + (-3.34 + 63.8i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(80.3−110.i)T+(−7.53e3−2.31e4i)T2 1 + (80.3 - 110. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(54.6−122.i)T+(−1.99e4−2.21e4i)T2 1 + (54.6 - 122. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−67.4+43.7i)T+(2.06e4−4.62e4i)T2 1 + (-67.4 + 43.7i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−157.+51.1i)T+(5.57e4−4.05e4i)T2 1 + (-157. + 51.1i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−52.0−52.0i)T+7.95e4iT2 1 + (-52.0 - 52.0i)T + 7.95e4iT^{2}
47 1+(−267.−102.i)T+(7.71e4+6.94e4i)T2 1 + (-267. - 102. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(499.−404.i)T+(3.09e4−1.45e5i)T2 1 + (499. - 404. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(558.−620.i)T+(−2.14e4−2.04e5i)T2 1 + (558. - 620. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−557.+502.i)T+(2.37e4−2.25e5i)T2 1 + (-557. + 502. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−698.+267.i)T+(2.23e5−2.01e5i)T2 1 + (-698. + 267. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−248.−180.i)T+(1.10e5+3.40e5i)T2 1 + (-248. - 180. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−360.+555.i)T+(−1.58e5−3.55e5i)T2 1 + (-360. + 555. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(−101.−228.i)T+(−3.29e5+3.66e5i)T2 1 + (-101. - 228. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−78.0+492.i)T+(−5.43e5−1.76e5i)T2 1 + (-78.0 + 492. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−377.−418.i)T+(−7.36e4+7.01e5i)T2 1 + (-377. - 418. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(−162.−1.02e3i)T+(−8.68e5+2.82e5i)T2 1 + (-162. - 1.02e3i)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

−12.37628694416241470685469434745, −10.76460717162131638566342325356, −10.60865819013033033106182012931, −9.327507681892167060814463366203, −8.167862948939658397168156907533, −7.50989212482119689751854217780, −6.18153574850963052580183672025, −5.21503192415873663035904075374, −2.20948018136201745352679186799, −1.27939372986389377334997821314, 0.53668226322172993228003092802, 2.33134835393061587407517604533, 4.86727097025416770603215244952, 5.54953959139182688569171464257, 7.33972276844449071330433969908, 8.103114279917202954054357866675, 9.518370233741221707922436139147, 9.959898965763092961500121620501, 10.92578153237046031411559900149, 11.35645698050762031672136719657

Graph of the ZZ-function along the critical line