↑

Properties

Label 2-832-13.11-c0-0-0
Degree 22
Conductor 832832
Sign 0.846−0.533i0.846 - 0.533i
Analytic cond. 0.4152220.415222
Root an. cond. 0.6443770.644377
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 1.36i)5-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + 2.73i·25-s + (−0.866 − 1.5i)29-s + (0.5 + 0.133i)37-s + (−0.133 + 0.5i)41-s + (1.86 − 0.499i)45-s + (0.866 − 0.5i)49-s − 53-s + (0.5 − 0.866i)61-s + (−1.86 − 0.499i)65-s + (0.366 − 0.366i)73-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)5-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + 2.73i·25-s + (−0.866 − 1.5i)29-s + (0.5 + 0.133i)37-s + (−0.133 + 0.5i)41-s + (1.86 − 0.499i)45-s + (0.866 − 0.5i)49-s − 53-s + (0.5 − 0.866i)61-s + (−1.86 − 0.499i)65-s + (0.366 − 0.366i)73-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.846−0.533i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s) L(s)=((0.846−0.533i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.846−0.533i0.846 - 0.533i
Analytic conductor: 0.4152220.415222
Root analytic conductor: 0.6443770.644377
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ832(193,⋅)\chi_{832} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :0), 0.846−0.533i)(2,\ 832,\ (\ :0),\ 0.846 - 0.533i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1882639811.188263981
L(12)L(\frac12) ≈\approx 1.1882639811.188263981
L(1)L(1) 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 1
13 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
good3 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
5 1+(−1.36−1.36i)T+iT2 1 + (-1.36 - 1.36i)T + iT^{2}
7 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
11 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
17 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
19 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
31 1+iT2 1 + iT^{2}
37 1+(−0.5−0.133i)T+(0.866+0.5i)T2 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2}
41 1+(0.133−0.5i)T+(−0.866−0.5i)T2 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1−iT2 1 - iT^{2}
53 1+T+T2 1 + T + T^{2}
59 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
61 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
71 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
73 1+(−0.366+0.366i)T−iT2 1 + (-0.366 + 0.366i)T - iT^{2}
79 1+T2 1 + T^{2}
83 1+iT2 1 + iT^{2}
89 1+(−1.36−0.366i)T+(0.866+0.5i)T2 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2}
97 1+(1.36−0.366i)T+(0.866−0.5i)T2 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)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

−10.37382200025768666036247918630, −9.466907280217285571772487614463, −9.406868570456088469415445568881, −7.71737650154475897965482258027, −6.76184675791351524618622552817, −6.42936583727978270483653683299, −5.35879540791534021062627480148, −4.06990739768071370138353787532, −2.81201065437481343476282945082, −1.96304313090749609697792757956, 1.52583611447171103101202627383, 2.44459611662714200620811490083, 4.30285106476210617640516640235, 5.11340395392256713975733595580, 5.69332215507483514729803568191, 6.86632471771616723842744582839, 7.927356117549501727336875750533, 8.812284877693319782250465107802, 9.459958437149121086492526454523, 10.24368090345367036235719720955

Graph of the ZZ-function along the critical line