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Properties

Label 2-2816-8.5-c1-0-54
Degree 22
Conductor 28162816
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 22.485822.4858
Root an. cond. 4.741924.74192
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − i·5-s − 6·9-s + i·11-s − 6i·13-s + 3·15-s − 4·17-s + 6i·19-s − 3·23-s + 4·25-s − 9i·27-s − 4i·29-s − 9·31-s − 3·33-s − 7i·37-s + ⋯
L(s)  = 1  + 1.73i·3-s − 0.447i·5-s − 2·9-s + 0.301i·11-s − 1.66i·13-s + 0.774·15-s − 0.970·17-s + 1.37i·19-s − 0.625·23-s + 0.800·25-s − 1.73i·27-s − 0.742i·29-s − 1.61·31-s − 0.522·33-s − 1.15i·37-s + ⋯

Functional equation

Λ(s)=(2816s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2816s/2 ΓC(s+1/2) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 28162816    =    28⋅112^{8} \cdot 11
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 22.485822.4858
Root analytic conductor: 4.741924.74192
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2816(1409,⋅)\chi_{2816} (1409, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2816, ( :1/2), 0.707+0.707i)(2,\ 2816,\ (\ :1/2),\ 0.707 + 0.707i)

Particular Values

L(1)L(1) ≈\approx 0.80940669420.8094066942
L(12)L(\frac12) ≈\approx 0.80940669420.8094066942
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
11 1−iT 1 - iT
good3 1−3iT−3T2 1 - 3iT - 3T^{2}
5 1+iT−5T2 1 + iT - 5T^{2}
7 1+7T2 1 + 7T^{2}
13 1+6iT−13T2 1 + 6iT - 13T^{2}
17 1+4T+17T2 1 + 4T + 17T^{2}
19 1−6iT−19T2 1 - 6iT - 19T^{2}
23 1+3T+23T2 1 + 3T + 23T^{2}
29 1+4iT−29T2 1 + 4iT - 29T^{2}
31 1+9T+31T2 1 + 9T + 31T^{2}
37 1+7iT−37T2 1 + 7iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+6iT−43T2 1 + 6iT - 43T^{2}
47 1−12T+47T2 1 - 12T + 47T^{2}
53 1+2iT−53T2 1 + 2iT - 53T^{2}
59 1+9iT−59T2 1 + 9iT - 59T^{2}
61 1−8iT−61T2 1 - 8iT - 61T^{2}
67 1+15iT−67T2 1 + 15iT - 67T^{2}
71 1−3T+71T2 1 - 3T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1+6T+79T2 1 + 6T + 79T^{2}
83 1+6iT−83T2 1 + 6iT - 83T^{2}
89 1−5T+89T2 1 - 5T + 89T^{2}
97 1+3T+97T2 1 + 3T + 97T^{2}
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   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

−8.828166532879313063266760358486, −8.203231858226989815633773288995, −7.37429782400931558499658528270, −5.95923148958372934368105614052, −5.51890285313055267382068267363, −4.73989642467808185576099402811, −3.94386681295991909761339123533, −3.35202094195077612571948163769, −2.15859539392913729547004501638, −0.26483538563779084555808931266, 1.20056245821313530318904163647, 2.16227545382332574653939063753, 2.82752987980947634028591631281, 4.11146394682894729592621045229, 5.16248964374009870609883321588, 6.25325784827471714775020220923, 6.77404503145119496374355093687, 7.11198342021762712135026152189, 7.970929161623730609973162627095, 8.965815256709578061755701176364

Graph of the ZZ-function along the critical line