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Properties

Label 2-1120-5.4-c1-0-11
Degree 22
Conductor 11201120
Sign 0.997+0.0685i0.997 + 0.0685i
Analytic cond. 8.943248.94324
Root an. cond. 2.990522.99052
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  − 1.63i·3-s + (−2.23 − 0.153i)5-s + i·7-s + 0.328·9-s − 1.24·11-s + 4.20i·13-s + (−0.250 + 3.64i)15-s + 3.39i·17-s + 6.46·19-s + 1.63·21-s + 2.15i·23-s + (4.95 + 0.683i)25-s − 5.44i·27-s − 3.96·29-s + 10.0·31-s + ⋯
L(s)  = 1  − 0.943i·3-s + (−0.997 − 0.0685i)5-s + 0.377i·7-s + 0.109·9-s − 0.374·11-s + 1.16i·13-s + (−0.0646 + 0.941i)15-s + 0.823i·17-s + 1.48·19-s + 0.356·21-s + 0.449i·23-s + (0.990 + 0.136i)25-s − 1.04i·27-s − 0.736·29-s + 1.80·31-s + ⋯

Functional equation

Λ(s)=(1120s/2 ΓC(s) L(s)=((0.997+0.0685i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0685i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1120s/2 ΓC(s+1/2) L(s)=((0.997+0.0685i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11201120    =    25⋅5⋅72^{5} \cdot 5 \cdot 7
Sign: 0.997+0.0685i0.997 + 0.0685i
Analytic conductor: 8.943248.94324
Root analytic conductor: 2.990522.99052
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1120(449,⋅)\chi_{1120} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1120, ( :1/2), 0.997+0.0685i)(2,\ 1120,\ (\ :1/2),\ 0.997 + 0.0685i)

Particular Values

L(1)L(1) ≈\approx 1.3335369211.333536921
L(12)L(\frac12) ≈\approx 1.3335369211.333536921
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
5 1+(2.23+0.153i)T 1 + (2.23 + 0.153i)T
7 1−iT 1 - iT
good3 1+1.63iT−3T2 1 + 1.63iT - 3T^{2}
11 1+1.24T+11T2 1 + 1.24T + 11T^{2}
13 1−4.20iT−13T2 1 - 4.20iT - 13T^{2}
17 1−3.39iT−17T2 1 - 3.39iT - 17T^{2}
19 1−6.46T+19T2 1 - 6.46T + 19T^{2}
23 1−2.15iT−23T2 1 - 2.15iT - 23T^{2}
29 1+3.96T+29T2 1 + 3.96T + 29T^{2}
31 1−10.0T+31T2 1 - 10.0T + 31T^{2}
37 1+6.76iT−37T2 1 + 6.76iT - 37T^{2}
41 1+0.131T+41T2 1 + 0.131T + 41T^{2}
43 1+7.40iT−43T2 1 + 7.40iT - 43T^{2}
47 1−4.82iT−47T2 1 - 4.82iT - 47T^{2}
53 1−10.0iT−53T2 1 - 10.0iT - 53T^{2}
59 1−10.9T+59T2 1 - 10.9T + 59T^{2}
61 1+6.33T+61T2 1 + 6.33T + 61T^{2}
67 1−2.65iT−67T2 1 - 2.65iT - 67T^{2}
71 1+0.754T+71T2 1 + 0.754T + 71T^{2}
73 1−6.03iT−73T2 1 - 6.03iT - 73T^{2}
79 1−14.6T+79T2 1 - 14.6T + 79T^{2}
83 1−14.0iT−83T2 1 - 14.0iT - 83T^{2}
89 1−12.6T+89T2 1 - 12.6T + 89T^{2}
97 1+0.914iT−97T2 1 + 0.914iT - 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

−9.719083030311344402702618136971, −8.861615607491800256126106358911, −7.964311054725694117851497855703, −7.41045137809122897598605450430, −6.69203538277729928321727426175, −5.68375639826399958964507200149, −4.53954529580680138688536212385, −3.61876102259087331397881670833, −2.31768480279854528430991147506, −1.08046439096713055611024755420, 0.76051904040828307288564385906, 2.96693950208420419442696808021, 3.58542445427792752040607136376, 4.72280397031164105487027960232, 5.15879105180176691363170630157, 6.58681468923159314117382433071, 7.58799012780162969530735674772, 8.039554746915273597197866426081, 9.144979207420570175852871518402, 10.05627704733567137109369978014

Graph of the ZZ-function along the critical line