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Properties

Label 2-175-175.103-c3-0-18
Degree 22
Conductor 175175
Sign 0.489−0.872i0.489 - 0.872i
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  + (−2.63 − 0.137i)2-s + (4.95 + 6.11i)3-s + (−1.04 − 0.110i)4-s + (0.357 − 11.1i)5-s + (−12.1 − 16.7i)6-s + (−18.4 − 0.896i)7-s + (23.5 + 3.73i)8-s + (−7.25 + 34.1i)9-s + (−2.48 + 29.3i)10-s + (53.7 − 11.4i)11-s + (−4.52 − 6.96i)12-s + (34.3 + 17.5i)13-s + (48.5 + 4.91i)14-s + (70.0 − 53.1i)15-s + (−53.2 − 11.3i)16-s + (25.2 + 65.6i)17-s + ⋯
L(s)  = 1  + (−0.930 − 0.0487i)2-s + (0.952 + 1.17i)3-s + (−0.131 − 0.0137i)4-s + (0.0319 − 0.999i)5-s + (−0.829 − 1.14i)6-s + (−0.998 − 0.0484i)7-s + (1.04 + 0.164i)8-s + (−0.268 + 1.26i)9-s + (−0.0784 + 0.928i)10-s + (1.47 − 0.312i)11-s + (−0.108 − 0.167i)12-s + (0.733 + 0.373i)13-s + (0.926 + 0.0937i)14-s + (1.20 − 0.914i)15-s + (−0.832 − 0.176i)16-s + (0.359 + 0.936i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.489−0.872i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.489−0.872i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.489−0.872i0.489 - 0.872i
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), 0.489−0.872i)(2,\ 175,\ (\ :3/2),\ 0.489 - 0.872i)

Particular Values

L(2)L(2) ≈\approx 1.09478+0.640969i1.09478 + 0.640969i
L(12)L(\frac12) ≈\approx 1.09478+0.640969i1.09478 + 0.640969i
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+(−0.357+11.1i)T 1 + (-0.357 + 11.1i)T
7 1+(18.4+0.896i)T 1 + (18.4 + 0.896i)T
good2 1+(2.63+0.137i)T+(7.95+0.836i)T2 1 + (2.63 + 0.137i)T + (7.95 + 0.836i)T^{2}
3 1+(−4.95−6.11i)T+(−5.61+26.4i)T2 1 + (-4.95 - 6.11i)T + (-5.61 + 26.4i)T^{2}
11 1+(−53.7+11.4i)T+(1.21e3−541.i)T2 1 + (-53.7 + 11.4i)T + (1.21e3 - 541. i)T^{2}
13 1+(−34.3−17.5i)T+(1.29e3+1.77e3i)T2 1 + (-34.3 - 17.5i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−25.2−65.6i)T+(−3.65e3+3.28e3i)T2 1 + (-25.2 - 65.6i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−1.12−10.6i)T+(−6.70e3+1.42e3i)T2 1 + (-1.12 - 10.6i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(9.13−174.i)T+(−1.21e4−1.27e3i)T2 1 + (9.13 - 174. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(−72.7+100.i)T+(−7.53e3−2.31e4i)T2 1 + (-72.7 + 100. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(72.1−161.i)T+(−1.99e4−2.21e4i)T2 1 + (72.1 - 161. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−145.+94.5i)T+(2.06e4−4.62e4i)T2 1 + (-145. + 94.5i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−228.+74.2i)T+(5.57e4−4.05e4i)T2 1 + (-228. + 74.2i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−259.−259.i)T+7.95e4iT2 1 + (-259. - 259. i)T + 7.95e4iT^{2}
47 1+(−150.−57.8i)T+(7.71e4+6.94e4i)T2 1 + (-150. - 57.8i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(361.−292.i)T+(3.09e4−1.45e5i)T2 1 + (361. - 292. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−104.+116.i)T+(−2.14e4−2.04e5i)T2 1 + (-104. + 116. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(667.−601.i)T+(2.37e4−2.25e5i)T2 1 + (667. - 601. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−935.+358.i)T+(2.23e5−2.01e5i)T2 1 + (-935. + 358. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(392.+285.i)T+(1.10e5+3.40e5i)T2 1 + (392. + 285. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−451.+695.i)T+(−1.58e5−3.55e5i)T2 1 + (-451. + 695. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(−332.−746.i)T+(−3.29e5+3.66e5i)T2 1 + (-332. - 746. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−121.+768.i)T+(−5.43e5−1.76e5i)T2 1 + (-121. + 768. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(323.+359.i)T+(−7.36e4+7.01e5i)T2 1 + (323. + 359. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(117.+740.i)T+(−8.68e5+2.82e5i)T2 1 + (117. + 740. 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

−12.45656141216963880305355408351, −11.01640454221087448566761613962, −9.867406968861939895246711865411, −9.225516763349434696172849658218, −8.916911822469795900650881186438, −7.83609116353689286760033135070, −5.99875089517887109492149830163, −4.30003833590056142548016369068, −3.63747474713698039572690684837, −1.28925142385483709197359397905, 0.875459045095436970796706456436, 2.50414113038047009225451903730, 3.80034789607156704017049658768, 6.42071411737072304896055561635, 7.01272861939209785622400755531, 7.990133013576499261913267217447, 9.035098700377352918420485961649, 9.693798670182584810641339454222, 10.88804388071032081765900903986, 12.26043869343087201004044650145

Graph of the ZZ-function along the critical line