↑

Properties

Label 2-171-19.7-c1-0-3
Degree 22
Conductor 171171
Sign 0.875+0.483i0.875 + 0.483i
Analytic cond. 1.365441.36544
Root an. cond. 1.168521.16852
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.370 − 0.642i)2-s + (0.724 − 1.25i)4-s + (1.65 + 2.85i)5-s + 1.44·7-s − 2.55·8-s + (1.22 − 2.12i)10-s + 1.81·11-s + (0.5 − 0.866i)13-s + (−0.537 − 0.931i)14-s + (−0.499 − 0.866i)16-s + (−3.30 − 5.71i)17-s + (1 + 4.24i)19-s + 4.78·20-s + (−0.674 − 1.16i)22-s + (2.39 − 4.14i)23-s + ⋯
L(s)  = 1  + (−0.262 − 0.454i)2-s + (0.362 − 0.627i)4-s + (0.738 + 1.27i)5-s + 0.547·7-s − 0.904·8-s + (0.387 − 0.670i)10-s + 0.547·11-s + (0.138 − 0.240i)13-s + (−0.143 − 0.248i)14-s + (−0.124 − 0.216i)16-s + (−0.800 − 1.38i)17-s + (0.229 + 0.973i)19-s + 1.07·20-s + (−0.143 − 0.248i)22-s + (0.498 − 0.864i)23-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((0.875+0.483i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+1/2) L(s)=((0.875+0.483i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: 0.875+0.483i0.875 + 0.483i
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ171(64,⋅)\chi_{171} (64, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 171, ( :1/2), 0.875+0.483i)(2,\ 171,\ (\ :1/2),\ 0.875 + 0.483i)

Particular Values

L(1)L(1) ≈\approx 1.19409−0.308067i1.19409 - 0.308067i
L(12)L(\frac12) ≈\approx 1.19409−0.308067i1.19409 - 0.308067i
L(32)L(\frac{3}{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)
bad3 1 1
19 1+(−1−4.24i)T 1 + (-1 - 4.24i)T
good2 1+(0.370+0.642i)T+(−1+1.73i)T2 1 + (0.370 + 0.642i)T + (-1 + 1.73i)T^{2}
5 1+(−1.65−2.85i)T+(−2.5+4.33i)T2 1 + (-1.65 - 2.85i)T + (-2.5 + 4.33i)T^{2}
7 1−1.44T+7T2 1 - 1.44T + 7T^{2}
11 1−1.81T+11T2 1 - 1.81T + 11T^{2}
13 1+(−0.5+0.866i)T+(−6.5−11.2i)T2 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2}
17 1+(3.30+5.71i)T+(−8.5+14.7i)T2 1 + (3.30 + 5.71i)T + (-8.5 + 14.7i)T^{2}
23 1+(−2.39+4.14i)T+(−11.5−19.9i)T2 1 + (-2.39 + 4.14i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.78−8.28i)T+(−14.5−25.1i)T2 1 + (4.78 - 8.28i)T + (-14.5 - 25.1i)T^{2}
31 1+4.55T+31T2 1 + 4.55T + 31T^{2}
37 1+5.89T+37T2 1 + 5.89T + 37T^{2}
41 1+(1.48+2.57i)T+(−20.5+35.5i)T2 1 + (1.48 + 2.57i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4.17−7.22i)T+(−21.5+37.2i)T2 1 + (-4.17 - 7.22i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.48+2.57i)T+(−23.5−40.7i)T2 1 + (-1.48 + 2.57i)T + (-23.5 - 40.7i)T^{2}
53 1+(1.65−2.85i)T+(−26.5−45.8i)T2 1 + (1.65 - 2.85i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.21+7.29i)T+(−29.5+51.0i)T2 1 + (4.21 + 7.29i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.5−4.33i)T+(−30.5−52.8i)T2 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2}
67 1+(−7.17+12.4i)T+(−33.5−58.0i)T2 1 + (-7.17 + 12.4i)T + (-33.5 - 58.0i)T^{2}
71 1+(−4.78−8.28i)T+(−35.5+61.4i)T2 1 + (-4.78 - 8.28i)T + (-35.5 + 61.4i)T^{2}
73 1+(2.5+4.33i)T+(−36.5+63.2i)T2 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2}
79 1+(−7.17−12.4i)T+(−39.5+68.4i)T2 1 + (-7.17 - 12.4i)T + (-39.5 + 68.4i)T^{2}
83 1+3.63T+83T2 1 + 3.63T + 83T^{2}
89 1+(−8.25+14.2i)T+(−44.5−77.0i)T2 1 + (-8.25 + 14.2i)T + (-44.5 - 77.0i)T^{2}
97 1+(6.44+11.1i)T+(−48.5+84.0i)T2 1 + (6.44 + 11.1i)T + (-48.5 + 84.0i)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

−12.48770230757098761678731620228, −11.20721183171400175481411858604, −10.87373539783600939498392576555, −9.846355497656934285992104771347, −8.971888544030087968661550512881, −7.21465936273925763301072800598, −6.41154737491163570596525498977, −5.24330761978999811517949769637, −3.13451600143324871986974396707, −1.85505864880742908792514435663, 1.87374757507543874498583370423, 4.01353707060061123016859263813, 5.40587237188096489899156098875, 6.53121004477192396556870709719, 7.81434876985752383896705223528, 8.832629028797874100892133635096, 9.323986813993517024613524527837, 11.00542243849056830576402305384, 11.92156717309070714521305227565, 12.92941540116239980597959495710

Graph of the ZZ-function along the critical line