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Properties

Label 2-175-175.103-c3-0-27
Degree 22
Conductor 175175
Sign 0.874+0.484i0.874 + 0.484i
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  + (−4.55 − 0.238i)2-s + (1.16 + 1.43i)3-s + (12.7 + 1.33i)4-s + (0.739 + 11.1i)5-s + (−4.94 − 6.80i)6-s + (4.38 − 17.9i)7-s + (−21.5 − 3.41i)8-s + (4.90 − 23.0i)9-s + (−0.705 − 50.9i)10-s + (−29.1 + 6.20i)11-s + (12.8 + 19.7i)12-s + (33.5 + 17.1i)13-s + (−24.2 + 80.8i)14-s + (−15.1 + 14.0i)15-s + (−2.73 − 0.582i)16-s + (−1.86 − 4.86i)17-s + ⋯
L(s)  = 1  + (−1.60 − 0.0843i)2-s + (0.223 + 0.276i)3-s + (1.58 + 0.167i)4-s + (0.0661 + 0.997i)5-s + (−0.336 − 0.463i)6-s + (0.236 − 0.971i)7-s + (−0.952 − 0.150i)8-s + (0.181 − 0.854i)9-s + (−0.0223 − 1.61i)10-s + (−0.799 + 0.169i)11-s + (0.309 + 0.476i)12-s + (0.716 + 0.365i)13-s + (−0.463 + 1.54i)14-s + (−0.260 + 0.241i)15-s + (−0.0427 − 0.00909i)16-s + (−0.0266 − 0.0694i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.874+0.484i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.874+0.484i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.874+0.484i0.874 + 0.484i
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), 0.874+0.484i)(2,\ 175,\ (\ :3/2),\ 0.874 + 0.484i)

Particular Values

L(2)L(2) ≈\approx 0.773784−0.199922i0.773784 - 0.199922i
L(12)L(\frac12) ≈\approx 0.773784−0.199922i0.773784 - 0.199922i
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+(−0.739−11.1i)T 1 + (-0.739 - 11.1i)T
7 1+(−4.38+17.9i)T 1 + (-4.38 + 17.9i)T
good2 1+(4.55+0.238i)T+(7.95+0.836i)T2 1 + (4.55 + 0.238i)T + (7.95 + 0.836i)T^{2}
3 1+(−1.16−1.43i)T+(−5.61+26.4i)T2 1 + (-1.16 - 1.43i)T + (-5.61 + 26.4i)T^{2}
11 1+(29.1−6.20i)T+(1.21e3−541.i)T2 1 + (29.1 - 6.20i)T + (1.21e3 - 541. i)T^{2}
13 1+(−33.5−17.1i)T+(1.29e3+1.77e3i)T2 1 + (-33.5 - 17.1i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(1.86+4.86i)T+(−3.65e3+3.28e3i)T2 1 + (1.86 + 4.86i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(10.3+98.7i)T+(−6.70e3+1.42e3i)T2 1 + (10.3 + 98.7i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(6.71−128.i)T+(−1.21e4−1.27e3i)T2 1 + (6.71 - 128. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(−148.+204.i)T+(−7.53e3−2.31e4i)T2 1 + (-148. + 204. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−48.2+108.i)T+(−1.99e4−2.21e4i)T2 1 + (-48.2 + 108. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−241.+156.i)T+(2.06e4−4.62e4i)T2 1 + (-241. + 156. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−373.+121.i)T+(5.57e4−4.05e4i)T2 1 + (-373. + 121. i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−257.−257.i)T+7.95e4iT2 1 + (-257. - 257. i)T + 7.95e4iT^{2}
47 1+(294.+113.i)T+(7.71e4+6.94e4i)T2 1 + (294. + 113. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(−293.+237.i)T+(3.09e4−1.45e5i)T2 1 + (-293. + 237. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(191.−212.i)T+(−2.14e4−2.04e5i)T2 1 + (191. - 212. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(36.1−32.5i)T+(2.37e4−2.25e5i)T2 1 + (36.1 - 32.5i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−571.+219.i)T+(2.23e5−2.01e5i)T2 1 + (-571. + 219. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−775.−563.i)T+(1.10e5+3.40e5i)T2 1 + (-775. - 563. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−86.5+133.i)T+(−1.58e5−3.55e5i)T2 1 + (-86.5 + 133. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(−148.−332.i)T+(−3.29e5+3.66e5i)T2 1 + (-148. - 332. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(104.−657.i)T+(−5.43e5−1.76e5i)T2 1 + (104. - 657. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(276.+306.i)T+(−7.36e4+7.01e5i)T2 1 + (276. + 306. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(−43.0−271.i)T+(−8.68e5+2.82e5i)T2 1 + (-43.0 - 271. 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

−11.43957067208015359747188238240, −10.97927059793786143775810120742, −9.958396378775007651552739227663, −9.416368753202821921371668050394, −8.078536189704356120073003709330, −7.28554598066365621324972362072, −6.35798461794319014463723027885, −4.09921729984198371867640362141, −2.54081732856392851211003981882, −0.72421104156829090653480537434, 1.16043952232493245501902253791, 2.41042978609625697956794561764, 4.90355788057364124277218907629, 6.18209073123685632363327291177, 7.85734340304876146258999701336, 8.286897878448972668125393492479, 8.989850687272822753700842153786, 10.20361206759916347285066494274, 10.95453723903529436002587836268, 12.29179480386280711632919021297

Graph of the ZZ-function along the critical line