↑

Properties

Label 2-234-117.110-c1-0-7
Degree 22
Conductor 234234
Sign 0.874−0.484i0.874 - 0.484i
Analytic cond. 1.868491.86849
Root an. cond. 1.366931.36693
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  + (−0.258 + 0.965i)2-s + (1.73 + 0.00498i)3-s + (−0.866 − 0.499i)4-s + (0.361 − 1.35i)5-s + (−0.453 + 1.67i)6-s + (−0.715 − 0.715i)7-s + (0.707 − 0.707i)8-s + (2.99 + 0.0172i)9-s + (1.21 + 0.699i)10-s + (5.61 + 1.50i)11-s + (−1.49 − 0.870i)12-s + (−0.218 + 3.59i)13-s + (0.876 − 0.505i)14-s + (0.633 − 2.33i)15-s + (0.500 + 0.866i)16-s + (−1.67 − 2.89i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.999 + 0.00287i)3-s + (−0.433 − 0.249i)4-s + (0.161 − 0.603i)5-s + (−0.184 + 0.682i)6-s + (−0.270 − 0.270i)7-s + (0.249 − 0.249i)8-s + (0.999 + 0.00575i)9-s + (0.382 + 0.221i)10-s + (1.69 + 0.453i)11-s + (−0.432 − 0.251i)12-s + (−0.0606 + 0.998i)13-s + (0.234 − 0.135i)14-s + (0.163 − 0.603i)15-s + (0.125 + 0.216i)16-s + (−0.405 − 0.702i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.874−0.484i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.874−0.484i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.874−0.484i0.874 - 0.484i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(227,⋅)\chi_{234} (227, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.874−0.484i)(2,\ 234,\ (\ :1/2),\ 0.874 - 0.484i)

Particular Values

L(1)L(1) ≈\approx 1.48559+0.383971i1.48559 + 0.383971i
L(12)L(\frac12) ≈\approx 1.48559+0.383971i1.48559 + 0.383971i
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)
bad2 1+(0.258−0.965i)T 1 + (0.258 - 0.965i)T
3 1+(−1.73−0.00498i)T 1 + (-1.73 - 0.00498i)T
13 1+(0.218−3.59i)T 1 + (0.218 - 3.59i)T
good5 1+(−0.361+1.35i)T+(−4.33−2.5i)T2 1 + (-0.361 + 1.35i)T + (-4.33 - 2.5i)T^{2}
7 1+(0.715+0.715i)T+7iT2 1 + (0.715 + 0.715i)T + 7iT^{2}
11 1+(−5.61−1.50i)T+(9.52+5.5i)T2 1 + (-5.61 - 1.50i)T + (9.52 + 5.5i)T^{2}
17 1+(1.67+2.89i)T+(−8.5+14.7i)T2 1 + (1.67 + 2.89i)T + (-8.5 + 14.7i)T^{2}
19 1+(7.07+1.89i)T+(16.4+9.5i)T2 1 + (7.07 + 1.89i)T + (16.4 + 9.5i)T^{2}
23 1−0.580T+23T2 1 - 0.580T + 23T^{2}
29 1+(0.892−0.515i)T+(14.5−25.1i)T2 1 + (0.892 - 0.515i)T + (14.5 - 25.1i)T^{2}
31 1+(8.52+2.28i)T+(26.8+15.5i)T2 1 + (8.52 + 2.28i)T + (26.8 + 15.5i)T^{2}
37 1+(7.82−2.09i)T+(32.0−18.5i)T2 1 + (7.82 - 2.09i)T + (32.0 - 18.5i)T^{2}
41 1+(−1.15−1.15i)T+41iT2 1 + (-1.15 - 1.15i)T + 41iT^{2}
43 1−8.38iT−43T2 1 - 8.38iT - 43T^{2}
47 1+(0.388+1.45i)T+(−40.7+23.5i)T2 1 + (0.388 + 1.45i)T + (-40.7 + 23.5i)T^{2}
53 1+2.77iT−53T2 1 + 2.77iT - 53T^{2}
59 1+(3.10+11.5i)T+(−51.0+29.5i)T2 1 + (3.10 + 11.5i)T + (-51.0 + 29.5i)T^{2}
61 1−2.82T+61T2 1 - 2.82T + 61T^{2}
67 1+(0.802−0.802i)T−67iT2 1 + (0.802 - 0.802i)T - 67iT^{2}
71 1+(2.71−10.1i)T+(−61.4−35.5i)T2 1 + (2.71 - 10.1i)T + (-61.4 - 35.5i)T^{2}
73 1+(−0.788−0.788i)T+73iT2 1 + (-0.788 - 0.788i)T + 73iT^{2}
79 1+(0.827−1.43i)T+(−39.5−68.4i)T2 1 + (0.827 - 1.43i)T + (-39.5 - 68.4i)T^{2}
83 1+(15.7−4.21i)T+(71.8−41.5i)T2 1 + (15.7 - 4.21i)T + (71.8 - 41.5i)T^{2}
89 1+(−0.783−2.92i)T+(−77.0+44.5i)T2 1 + (-0.783 - 2.92i)T + (-77.0 + 44.5i)T^{2}
97 1+(−8.70+8.70i)T−97iT2 1 + (-8.70 + 8.70i)T - 97iT^{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

−12.60605647427349043518487113859, −11.27612590264436316957825764429, −9.786762905548961121065863479862, −9.096384709656616514597186108496, −8.613098746648845220570519884187, −7.08911508935083289746505175869, −6.60992190698929842234354519452, −4.72615966995407827330159281353, −3.86660223023948450224631339518, −1.75921381894034822942112914357, 1.87285061981644706703732569124, 3.23827690610814825183028813746, 4.11112295337564417213465391192, 6.10019511013137214319282695210, 7.20613461967304763150689594377, 8.643974006315674513085362176038, 8.987101739807531915825708032602, 10.29477723473473475366689341039, 10.85230268152106656920112700343, 12.29535128930376558868813928854

Graph of the ZZ-function along the critical line