↑

Properties

Label 2-175-175.17-c3-0-49
Degree 22
Conductor 175175
Sign −0.999−0.0373i-0.999 - 0.0373i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.92 + 0.257i)2-s + (6.01 − 7.43i)3-s + (16.2 − 1.70i)4-s + (10.4 − 4.02i)5-s + (−27.7 + 38.1i)6-s + (−17.0 + 7.18i)7-s + (−40.4 + 6.40i)8-s + (−13.3 − 63.0i)9-s + (−50.3 + 22.4i)10-s + (−36.2 − 7.69i)11-s + (84.8 − 130. i)12-s + (−55.0 + 28.0i)13-s + (82.1 − 39.7i)14-s + (32.8 − 101. i)15-s + (69.7 − 14.8i)16-s + (−22.1 + 57.7i)17-s + ⋯
L(s)  = 1  + (−1.74 + 0.0912i)2-s + (1.15 − 1.43i)3-s + (2.02 − 0.212i)4-s + (0.933 − 0.359i)5-s + (−1.88 + 2.59i)6-s + (−0.921 + 0.388i)7-s + (−1.78 + 0.282i)8-s + (−0.496 − 2.33i)9-s + (−1.59 + 0.711i)10-s + (−0.992 − 0.210i)11-s + (2.04 − 3.14i)12-s + (−1.17 + 0.598i)13-s + (1.56 − 0.759i)14-s + (0.566 − 1.75i)15-s + (1.08 − 0.231i)16-s + (−0.316 + 0.823i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.999−0.0373i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0373i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.999−0.0373i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0373i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.999−0.0373i-0.999 - 0.0373i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(17,⋅)\chi_{175} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), −0.999−0.0373i)(2,\ 175,\ (\ :3/2),\ -0.999 - 0.0373i)

Particular Values

L(2)L(2) ≈\approx 0.0118615+0.634902i0.0118615 + 0.634902i
L(12)L(\frac12) ≈\approx 0.0118615+0.634902i0.0118615 + 0.634902i
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+(−10.4+4.02i)T 1 + (-10.4 + 4.02i)T
7 1+(17.0−7.18i)T 1 + (17.0 - 7.18i)T
good2 1+(4.92−0.257i)T+(7.95−0.836i)T2 1 + (4.92 - 0.257i)T + (7.95 - 0.836i)T^{2}
3 1+(−6.01+7.43i)T+(−5.61−26.4i)T2 1 + (-6.01 + 7.43i)T + (-5.61 - 26.4i)T^{2}
11 1+(36.2+7.69i)T+(1.21e3+541.i)T2 1 + (36.2 + 7.69i)T + (1.21e3 + 541. i)T^{2}
13 1+(55.0−28.0i)T+(1.29e3−1.77e3i)T2 1 + (55.0 - 28.0i)T + (1.29e3 - 1.77e3i)T^{2}
17 1+(22.1−57.7i)T+(−3.65e3−3.28e3i)T2 1 + (22.1 - 57.7i)T + (-3.65e3 - 3.28e3i)T^{2}
19 1+(−10.8+103.i)T+(−6.70e3−1.42e3i)T2 1 + (-10.8 + 103. i)T + (-6.70e3 - 1.42e3i)T^{2}
23 1+(−1.82−34.8i)T+(−1.21e4+1.27e3i)T2 1 + (-1.82 - 34.8i)T + (-1.21e4 + 1.27e3i)T^{2}
29 1+(31.4+43.3i)T+(−7.53e3+2.31e4i)T2 1 + (31.4 + 43.3i)T + (-7.53e3 + 2.31e4i)T^{2}
31 1+(118.+267.i)T+(−1.99e4+2.21e4i)T2 1 + (118. + 267. i)T + (-1.99e4 + 2.21e4i)T^{2}
37 1+(39.8+25.8i)T+(2.06e4+4.62e4i)T2 1 + (39.8 + 25.8i)T + (2.06e4 + 4.62e4i)T^{2}
41 1+(−33.6−10.9i)T+(5.57e4+4.05e4i)T2 1 + (-33.6 - 10.9i)T + (5.57e4 + 4.05e4i)T^{2}
43 1+(−155.+155.i)T−7.95e4iT2 1 + (-155. + 155. i)T - 7.95e4iT^{2}
47 1+(161.−62.1i)T+(7.71e4−6.94e4i)T2 1 + (161. - 62.1i)T + (7.71e4 - 6.94e4i)T^{2}
53 1+(−33.8−27.3i)T+(3.09e4+1.45e5i)T2 1 + (-33.8 - 27.3i)T + (3.09e4 + 1.45e5i)T^{2}
59 1+(−270.−300.i)T+(−2.14e4+2.04e5i)T2 1 + (-270. - 300. i)T + (-2.14e4 + 2.04e5i)T^{2}
61 1+(−323.−290.i)T+(2.37e4+2.25e5i)T2 1 + (-323. - 290. i)T + (2.37e4 + 2.25e5i)T^{2}
67 1+(−161.−61.9i)T+(2.23e5+2.01e5i)T2 1 + (-161. - 61.9i)T + (2.23e5 + 2.01e5i)T^{2}
71 1+(258.−187.i)T+(1.10e5−3.40e5i)T2 1 + (258. - 187. i)T + (1.10e5 - 3.40e5i)T^{2}
73 1+(426.+657.i)T+(−1.58e5+3.55e5i)T2 1 + (426. + 657. i)T + (-1.58e5 + 3.55e5i)T^{2}
79 1+(−350.+786.i)T+(−3.29e5−3.66e5i)T2 1 + (-350. + 786. i)T + (-3.29e5 - 3.66e5i)T^{2}
83 1+(27.2+172.i)T+(−5.43e5+1.76e5i)T2 1 + (27.2 + 172. i)T + (-5.43e5 + 1.76e5i)T^{2}
89 1+(−426.+473.i)T+(−7.36e4−7.01e5i)T2 1 + (-426. + 473. i)T + (-7.36e4 - 7.01e5i)T^{2}
97 1+(−234.+1.48e3i)T+(−8.68e5−2.82e5i)T2 1 + (-234. + 1.48e3i)T + (-8.68e5 - 2.82e5i)T^{2}
show more
show less
   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

−11.75746244781932107397343265225, −10.20128145833379688138927944342, −9.257719145812341874294388622862, −8.891611777615834784530473623380, −7.73559375962770525579563273578, −6.98586516657973673310936040685, −6.01031221353963607827422799172, −2.61594902929862288685182932213, −2.03734825586936992995347149462, −0.39178014286763846273097831274, 2.33033921624057747447532650367, 3.16248654212963464212977112626, 5.20485423768531611588139923541, 7.04301529369881375847089702964, 8.004835651401623457978948376805, 9.101532314694259653586075702127, 9.832382948607493089554442896044, 10.17256445105658024789138433071, 10.83030494746350303286683873103, 12.75475150763315007551139933398

Graph of the ZZ-function along the critical line