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Properties

Label 2-33e2-11.10-c2-0-43
Degree 22
Conductor 10891089
Sign 0.372−0.927i0.372 - 0.927i
Analytic cond. 29.673129.6731
Root an. cond. 5.447305.44730
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.07i·2-s − 5.47·4-s − 4·5-s + 0.898i·7-s − 4.53i·8-s − 12.3i·10-s − 8.50i·13-s − 2.76·14-s − 7.94·16-s − 24.7i·17-s + 11.8i·19-s + 21.8·20-s + 7.23·23-s − 9·25-s + 26.1·26-s + ⋯
L(s)  = 1  + 1.53i·2-s − 1.36·4-s − 0.800·5-s + 0.128i·7-s − 0.566i·8-s − 1.23i·10-s − 0.654i·13-s − 0.197·14-s − 0.496·16-s − 1.45i·17-s + 0.624i·19-s + 1.09·20-s + 0.314·23-s − 0.359·25-s + 1.00·26-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((0.372−0.927i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+1) L(s)=((0.372−0.927i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.372 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: 0.372−0.927i0.372 - 0.927i
Analytic conductor: 29.673129.6731
Root analytic conductor: 5.447305.44730
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ1089(604,⋅)\chi_{1089} (604, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1089, ( :1), 0.372−0.927i)(2,\ 1089,\ (\ :1),\ 0.372 - 0.927i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.2434653451.243465345
L(12)L(\frac12) ≈\approx 1.2434653451.243465345
L(2)L(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)
bad3 1 1
11 1 1
good2 1−3.07iT−4T2 1 - 3.07iT - 4T^{2}
5 1+4T+25T2 1 + 4T + 25T^{2}
7 1−0.898iT−49T2 1 - 0.898iT - 49T^{2}
13 1+8.50iT−169T2 1 + 8.50iT - 169T^{2}
17 1+24.7iT−289T2 1 + 24.7iT - 289T^{2}
19 1−11.8iT−361T2 1 - 11.8iT - 361T^{2}
23 1−7.23T+529T2 1 - 7.23T + 529T^{2}
29 1+3.46iT−841T2 1 + 3.46iT - 841T^{2}
31 1−33.1T+961T2 1 - 33.1T + 961T^{2}
37 1−40.2T+1.36e3T2 1 - 40.2T + 1.36e3T^{2}
41 1+1.30iT−1.68e3T2 1 + 1.30iT - 1.68e3T^{2}
43 1+33.0iT−1.84e3T2 1 + 33.0iT - 1.84e3T^{2}
47 1+22.7T+2.20e3T2 1 + 22.7T + 2.20e3T^{2}
53 1−78.5T+2.80e3T2 1 - 78.5T + 2.80e3T^{2}
59 1+31.2T+3.48e3T2 1 + 31.2T + 3.48e3T^{2}
61 1−28.0iT−3.72e3T2 1 - 28.0iT - 3.72e3T^{2}
67 1+76.5T+4.48e3T2 1 + 76.5T + 4.48e3T^{2}
71 1−62.3T+5.04e3T2 1 - 62.3T + 5.04e3T^{2}
73 1+94.1iT−5.32e3T2 1 + 94.1iT - 5.32e3T^{2}
79 1−65.9iT−6.24e3T2 1 - 65.9iT - 6.24e3T^{2}
83 1−56.7iT−6.88e3T2 1 - 56.7iT - 6.88e3T^{2}
89 1−62.2T+7.92e3T2 1 - 62.2T + 7.92e3T^{2}
97 1−72.4T+9.40e3T2 1 - 72.4T + 9.40e3T^{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

−9.501596950450486762150612086219, −8.703751413638769215744651275086, −7.86461139792563280408664680456, −7.47225918144445960541982289732, −6.56901894183215736128440586844, −5.67311910389124046665553135231, −4.90423325069693016767500345821, −3.96483852507239753342913204407, −2.66452834322818774315286715840, −0.53237918866483149563176950493, 0.863041628387778892829239717971, 2.04959420830303774468371384474, 3.14531416173286865224890021001, 4.06411913434107265078866979670, 4.59741990788380711584796739462, 6.09103962341931874258136036438, 7.10303084447568804025431362874, 8.140039777066700742460387297728, 8.893361858892043576505057759912, 9.759960624330986994180541216267

Graph of the ZZ-function along the critical line