↑

Properties

Label 2-3276-13.10-c1-0-0
Degree 22
Conductor 32763276
Sign −0.998−0.0599i-0.998 - 0.0599i
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
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  + 1.38i·5-s + (0.866 + 0.5i)7-s + (−0.998 + 0.576i)11-s + (−1.65 − 3.20i)13-s + (0.781 − 1.35i)17-s + (−7.26 − 4.19i)19-s + (1.11 + 1.92i)23-s + 3.08·25-s + (1.43 + 2.48i)29-s + 6.65i·31-s + (−0.692 + 1.19i)35-s + (3.01 − 1.74i)37-s + (−10.8 + 6.27i)41-s + (−3.07 + 5.33i)43-s + 7.40i·47-s + ⋯
L(s)  = 1  + 0.619i·5-s + (0.327 + 0.188i)7-s + (−0.301 + 0.173i)11-s + (−0.458 − 0.888i)13-s + (0.189 − 0.328i)17-s + (−1.66 − 0.962i)19-s + (0.231 + 0.401i)23-s + 0.616·25-s + (0.266 + 0.461i)29-s + 1.19i·31-s + (−0.117 + 0.202i)35-s + (0.495 − 0.286i)37-s + (−1.69 + 0.980i)41-s + (−0.469 + 0.813i)43-s + 1.08i·47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((−0.998−0.0599i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((−0.998−0.0599i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: −0.998−0.0599i-0.998 - 0.0599i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(1765,⋅)\chi_{3276} (1765, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), −0.998−0.0599i)(2,\ 3276,\ (\ :1/2),\ -0.998 - 0.0599i)

Particular Values

L(1)L(1) ≈\approx 0.31350945430.3135094543
L(12)L(\frac12) ≈\approx 0.31350945430.3135094543
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 1
3 1 1
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1+(1.65+3.20i)T 1 + (1.65 + 3.20i)T
good5 1−1.38iT−5T2 1 - 1.38iT - 5T^{2}
11 1+(0.998−0.576i)T+(5.5−9.52i)T2 1 + (0.998 - 0.576i)T + (5.5 - 9.52i)T^{2}
17 1+(−0.781+1.35i)T+(−8.5−14.7i)T2 1 + (-0.781 + 1.35i)T + (-8.5 - 14.7i)T^{2}
19 1+(7.26+4.19i)T+(9.5+16.4i)T2 1 + (7.26 + 4.19i)T + (9.5 + 16.4i)T^{2}
23 1+(−1.11−1.92i)T+(−11.5+19.9i)T2 1 + (-1.11 - 1.92i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.43−2.48i)T+(−14.5+25.1i)T2 1 + (-1.43 - 2.48i)T + (-14.5 + 25.1i)T^{2}
31 1−6.65iT−31T2 1 - 6.65iT - 31T^{2}
37 1+(−3.01+1.74i)T+(18.5−32.0i)T2 1 + (-3.01 + 1.74i)T + (18.5 - 32.0i)T^{2}
41 1+(10.8−6.27i)T+(20.5−35.5i)T2 1 + (10.8 - 6.27i)T + (20.5 - 35.5i)T^{2}
43 1+(3.07−5.33i)T+(−21.5−37.2i)T2 1 + (3.07 - 5.33i)T + (-21.5 - 37.2i)T^{2}
47 1−7.40iT−47T2 1 - 7.40iT - 47T^{2}
53 1+11.2T+53T2 1 + 11.2T + 53T^{2}
59 1+(6.97+4.02i)T+(29.5+51.0i)T2 1 + (6.97 + 4.02i)T + (29.5 + 51.0i)T^{2}
61 1+(0.829−1.43i)T+(−30.5−52.8i)T2 1 + (0.829 - 1.43i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.21−1.85i)T+(33.5−58.0i)T2 1 + (3.21 - 1.85i)T + (33.5 - 58.0i)T^{2}
71 1+(8.60+4.96i)T+(35.5+61.4i)T2 1 + (8.60 + 4.96i)T + (35.5 + 61.4i)T^{2}
73 1+3.30iT−73T2 1 + 3.30iT - 73T^{2}
79 1−4.64T+79T2 1 - 4.64T + 79T^{2}
83 1+1.82iT−83T2 1 + 1.82iT - 83T^{2}
89 1+(2.05−1.18i)T+(44.5−77.0i)T2 1 + (2.05 - 1.18i)T + (44.5 - 77.0i)T^{2}
97 1+(6.53+3.77i)T+(48.5+84.0i)T2 1 + (6.53 + 3.77i)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

−8.926767644387829563726759054751, −8.239978770492589758561182778491, −7.52101324353806833253211088844, −6.75275297428382288283162335305, −6.14607067665516878720371293935, −4.96911154923371773676514885196, −4.67200401978827975416721952345, −3.18146139637107332950062514206, −2.76328988808879524712243239636, −1.53199388859102603460161234296, 0.090358064393928066624774632116, 1.56150463914353849337620011411, 2.37735660485532666024508841213, 3.71183574604721569907754868601, 4.43635139730649648996983957604, 5.08560788182521583842170104695, 6.06912843886249595157109268313, 6.71840576511552826100738118483, 7.64688395988022864876898871930, 8.422787289343644551864277725347

Graph of the ZZ-function along the critical line