↑

Properties

Label 2-171-171.14-c1-0-4
Degree 22
Conductor 171171
Sign −0.496+0.868i-0.496 + 0.868i
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  + (−1.86 − 1.56i)2-s + (−1.40 + 1.01i)3-s + (0.684 + 3.87i)4-s + (0.0603 + 0.165i)5-s + (4.21 + 0.303i)6-s + (0.340 − 0.588i)7-s + (2.36 − 4.09i)8-s + (0.939 − 2.84i)9-s + (0.146 − 0.403i)10-s − 5.07i·11-s + (−4.89 − 4.75i)12-s + (−0.848 + 2.33i)13-s + (−1.55 + 0.566i)14-s + (−0.252 − 0.171i)15-s + (−3.41 + 1.24i)16-s + (−1.07 − 2.95i)17-s + ⋯
L(s)  = 1  + (−1.32 − 1.10i)2-s + (−0.810 + 0.586i)3-s + (0.342 + 1.93i)4-s + (0.0269 + 0.0741i)5-s + (1.71 + 0.123i)6-s + (0.128 − 0.222i)7-s + (0.835 − 1.44i)8-s + (0.313 − 0.949i)9-s + (0.0464 − 0.127i)10-s − 1.52i·11-s + (−1.41 − 1.37i)12-s + (−0.235 + 0.646i)13-s + (−0.416 + 0.151i)14-s + (−0.0652 − 0.0442i)15-s + (−0.854 + 0.311i)16-s + (−0.261 − 0.717i)17-s + ⋯

Functional equation

Λ(s)=(171s/2 ΓC(s) L(s)=((−0.496+0.868i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(171s/2 ΓC(s+1/2) L(s)=((−0.496+0.868i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 171171    =    32⋅193^{2} \cdot 19
Sign: −0.496+0.868i-0.496 + 0.868i
Analytic conductor: 1.365441.36544
Root analytic conductor: 1.168521.16852
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ171(14,⋅)\chi_{171} (14, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 171, ( :1/2), −0.496+0.868i)(2,\ 171,\ (\ :1/2),\ -0.496 + 0.868i)

Particular Values

L(1)L(1) ≈\approx 0.193923−0.334122i0.193923 - 0.334122i
L(12)L(\frac12) ≈\approx 0.193923−0.334122i0.193923 - 0.334122i
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+(1.40−1.01i)T 1 + (1.40 - 1.01i)T
19 1+(0.122+4.35i)T 1 + (0.122 + 4.35i)T
good2 1+(1.86+1.56i)T+(0.347+1.96i)T2 1 + (1.86 + 1.56i)T + (0.347 + 1.96i)T^{2}
5 1+(−0.0603−0.165i)T+(−3.83+3.21i)T2 1 + (-0.0603 - 0.165i)T + (-3.83 + 3.21i)T^{2}
7 1+(−0.340+0.588i)T+(−3.5−6.06i)T2 1 + (-0.340 + 0.588i)T + (-3.5 - 6.06i)T^{2}
11 1+5.07iT−11T2 1 + 5.07iT - 11T^{2}
13 1+(0.848−2.33i)T+(−9.95−8.35i)T2 1 + (0.848 - 2.33i)T + (-9.95 - 8.35i)T^{2}
17 1+(1.07+2.95i)T+(−13.0+10.9i)T2 1 + (1.07 + 2.95i)T + (-13.0 + 10.9i)T^{2}
23 1+(0.413−0.0728i)T+(21.6−7.86i)T2 1 + (0.413 - 0.0728i)T + (21.6 - 7.86i)T^{2}
29 1+(−0.0433−0.245i)T+(−27.2+9.91i)T2 1 + (-0.0433 - 0.245i)T + (-27.2 + 9.91i)T^{2}
31 1+9.69iT−31T2 1 + 9.69iT - 31T^{2}
37 1+5.05iT−37T2 1 + 5.05iT - 37T^{2}
41 1+(−8.91−7.48i)T+(7.11+40.3i)T2 1 + (-8.91 - 7.48i)T + (7.11 + 40.3i)T^{2}
43 1+(−0.329+1.86i)T+(−40.4−14.7i)T2 1 + (-0.329 + 1.86i)T + (-40.4 - 14.7i)T^{2}
47 1+(−0.352+0.0621i)T+(44.1−16.0i)T2 1 + (-0.352 + 0.0621i)T + (44.1 - 16.0i)T^{2}
53 1+(4.97−4.17i)T+(9.20−52.1i)T2 1 + (4.97 - 4.17i)T + (9.20 - 52.1i)T^{2}
59 1+(0.574−3.25i)T+(−55.4−20.1i)T2 1 + (0.574 - 3.25i)T + (-55.4 - 20.1i)T^{2}
61 1+(7.26+2.64i)T+(46.7+39.2i)T2 1 + (7.26 + 2.64i)T + (46.7 + 39.2i)T^{2}
67 1+(4.89+5.83i)T+(−11.6+65.9i)T2 1 + (4.89 + 5.83i)T + (-11.6 + 65.9i)T^{2}
71 1+(10.2+8.56i)T+(12.3+69.9i)T2 1 + (10.2 + 8.56i)T + (12.3 + 69.9i)T^{2}
73 1+(2.55−14.4i)T+(−68.5−24.9i)T2 1 + (2.55 - 14.4i)T + (-68.5 - 24.9i)T^{2}
79 1+(−2.04−5.61i)T+(−60.5+50.7i)T2 1 + (-2.04 - 5.61i)T + (-60.5 + 50.7i)T^{2}
83 1+(0.844+0.487i)T+(41.5+71.8i)T2 1 + (0.844 + 0.487i)T + (41.5 + 71.8i)T^{2}
89 1+(1.48+8.40i)T+(−83.6+30.4i)T2 1 + (1.48 + 8.40i)T + (-83.6 + 30.4i)T^{2}
97 1+(−8.73+10.4i)T+(−16.8−95.5i)T2 1 + (-8.73 + 10.4i)T + (-16.8 - 95.5i)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

−11.78144839240413789584676646943, −11.19997961892020366922926702875, −10.64635962347737243580488985825, −9.453260563393060053447497589117, −8.899395922339049933060582647526, −7.51264850693446009845468795418, −6.09475660967961523334554216699, −4.37235876670099440361746848143, −2.85959993701283800515640726211, −0.63801981691009384345949693835, 1.61202404851202775768854012557, 4.95085069437632087397836249225, 6.01709945459651121279465996065, 7.04639421125818055735995652675, 7.74159969131299684694385246220, 8.811734052246018755783881052994, 10.08518936812536544329927555064, 10.63978454178989736008884830713, 12.15110566172979926152007742560, 12.85504131036455083538718259039

Graph of the ZZ-function along the critical line