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Properties

Label 2-175-175.103-c3-0-12
Degree 22
Conductor 175175
Sign 0.947+0.320i0.947 + 0.320i
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  + (−1.59 − 0.0837i)2-s + (−4.11 − 5.08i)3-s + (−5.40 − 0.568i)4-s + (−9.89 + 5.20i)5-s + (6.15 + 8.46i)6-s + (−14.4 + 11.5i)7-s + (21.2 + 3.36i)8-s + (−3.27 + 15.3i)9-s + (16.2 − 7.49i)10-s + (−56.8 + 12.0i)11-s + (19.3 + 29.8i)12-s + (−10.5 − 5.36i)13-s + (24.0 − 17.2i)14-s + (67.1 + 28.8i)15-s + (8.85 + 1.88i)16-s + (−7.45 − 19.4i)17-s + ⋯
L(s)  = 1  + (−0.565 − 0.0296i)2-s + (−0.791 − 0.977i)3-s + (−0.675 − 0.0710i)4-s + (−0.884 + 0.465i)5-s + (0.418 + 0.576i)6-s + (−0.781 + 0.624i)7-s + (0.939 + 0.148i)8-s + (−0.121 + 0.570i)9-s + (0.514 − 0.236i)10-s + (−1.55 + 0.331i)11-s + (0.465 + 0.717i)12-s + (−0.224 − 0.114i)13-s + (0.460 − 0.329i)14-s + (1.15 + 0.496i)15-s + (0.138 + 0.0294i)16-s + (−0.106 − 0.276i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.947+0.320i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.947+0.320i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.947+0.320i0.947 + 0.320i
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), 0.947+0.320i)(2,\ 175,\ (\ :3/2),\ 0.947 + 0.320i)

Particular Values

L(2)L(2) ≈\approx 0.255809−0.0421656i0.255809 - 0.0421656i
L(12)L(\frac12) ≈\approx 0.255809−0.0421656i0.255809 - 0.0421656i
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+(9.89−5.20i)T 1 + (9.89 - 5.20i)T
7 1+(14.4−11.5i)T 1 + (14.4 - 11.5i)T
good2 1+(1.59+0.0837i)T+(7.95+0.836i)T2 1 + (1.59 + 0.0837i)T + (7.95 + 0.836i)T^{2}
3 1+(4.11+5.08i)T+(−5.61+26.4i)T2 1 + (4.11 + 5.08i)T + (-5.61 + 26.4i)T^{2}
11 1+(56.8−12.0i)T+(1.21e3−541.i)T2 1 + (56.8 - 12.0i)T + (1.21e3 - 541. i)T^{2}
13 1+(10.5+5.36i)T+(1.29e3+1.77e3i)T2 1 + (10.5 + 5.36i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(7.45+19.4i)T+(−3.65e3+3.28e3i)T2 1 + (7.45 + 19.4i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(8.62+82.0i)T+(−6.70e3+1.42e3i)T2 1 + (8.62 + 82.0i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(0.885−16.8i)T+(−1.21e4−1.27e3i)T2 1 + (0.885 - 16.8i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(−93.2+128.i)T+(−7.53e3−2.31e4i)T2 1 + (-93.2 + 128. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−2.66+5.98i)T+(−1.99e4−2.21e4i)T2 1 + (-2.66 + 5.98i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(210.−136.i)T+(2.06e4−4.62e4i)T2 1 + (210. - 136. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−262.+85.2i)T+(5.57e4−4.05e4i)T2 1 + (-262. + 85.2i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−313.−313.i)T+7.95e4iT2 1 + (-313. - 313. i)T + 7.95e4iT^{2}
47 1+(−305.−117.i)T+(7.71e4+6.94e4i)T2 1 + (-305. - 117. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(371.−301.i)T+(3.09e4−1.45e5i)T2 1 + (371. - 301. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(376.−418.i)T+(−2.14e4−2.04e5i)T2 1 + (376. - 418. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−32.2+29.0i)T+(2.37e4−2.25e5i)T2 1 + (-32.2 + 29.0i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(160.−61.6i)T+(2.23e5−2.01e5i)T2 1 + (160. - 61.6i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(212.+154.i)T+(1.10e5+3.40e5i)T2 1 + (212. + 154. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−332.+511.i)T+(−1.58e5−3.55e5i)T2 1 + (-332. + 511. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(105.+237.i)T+(−3.29e5+3.66e5i)T2 1 + (105. + 237. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(200.−1.26e3i)T+(−5.43e5−1.76e5i)T2 1 + (200. - 1.26e3i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(621.+690.i)T+(−7.36e4+7.01e5i)T2 1 + (621. + 690. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(160.+1.01e3i)T+(−8.68e5+2.82e5i)T2 1 + (160. + 1.01e3i)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

−12.34123963695066235573412818410, −11.19990432308991643304970278270, −10.29071944958069824844962003962, −9.128526588754261880779683482729, −7.87776821303969113251561776373, −7.20895560871345278265534045760, −5.93520802773859395724745251533, −4.66569112503282508364772755407, −2.72057945850257021327463892021, −0.48069041928634802393870108664, 0.39413555068805695081450345532, 3.64082799771803796745804437039, 4.59074842869579561965501993061, 5.55826495567651534317814965535, 7.39401616703320198923690918842, 8.297767837102523137437634650557, 9.433310023492391645976565993054, 10.48510764659891912629988288712, 10.75255844073591600161194335430, 12.34174495907763076852519744354

Graph of the ZZ-function along the critical line