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Properties

Label 2-1323-21.20-c1-0-12
Degree 22
Conductor 13231323
Sign 0.755+0.654i0.755 + 0.654i
Analytic cond. 10.564210.5642
Root an. cond. 3.250263.25026
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  − 1.83i·2-s − 1.36·4-s − 3.80·5-s − 1.16i·8-s + 6.98i·10-s + 1.16i·11-s + 5.54i·13-s − 4.86·16-s + 3.17·17-s − 0.631i·19-s + 5.19·20-s + 2.13·22-s + 1.76i·23-s + 9.50·25-s + 10.1·26-s + ⋯
L(s)  = 1  − 1.29i·2-s − 0.682·4-s − 1.70·5-s − 0.412i·8-s + 2.20i·10-s + 0.351i·11-s + 1.53i·13-s − 1.21·16-s + 0.770·17-s − 0.144i·19-s + 1.16·20-s + 0.455·22-s + 0.367i·23-s + 1.90·25-s + 1.99·26-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((0.755+0.654i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=((0.755+0.654i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 0.755+0.654i0.755 + 0.654i
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1323(1322,⋅)\chi_{1323} (1322, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), 0.755+0.654i)(2,\ 1323,\ (\ :1/2),\ 0.755 + 0.654i)

Particular Values

L(1)L(1) ≈\approx 1.0248121261.024812126
L(12)L(\frac12) ≈\approx 1.0248121261.024812126
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)
bad3 1 1
7 1 1
good2 1+1.83iT−2T2 1 + 1.83iT - 2T^{2}
5 1+3.80T+5T2 1 + 3.80T + 5T^{2}
11 1−1.16iT−11T2 1 - 1.16iT - 11T^{2}
13 1−5.54iT−13T2 1 - 5.54iT - 13T^{2}
17 1−3.17T+17T2 1 - 3.17T + 17T^{2}
19 1+0.631iT−19T2 1 + 0.631iT - 19T^{2}
23 1−1.76iT−23T2 1 - 1.76iT - 23T^{2}
29 1+4.83iT−29T2 1 + 4.83iT - 29T^{2}
31 1−4.27iT−31T2 1 - 4.27iT - 31T^{2}
37 1−4.23T+37T2 1 - 4.23T + 37T^{2}
41 1−4.56T+41T2 1 - 4.56T + 41T^{2}
43 1−7.23T+43T2 1 - 7.23T + 43T^{2}
47 1+2.54T+47T2 1 + 2.54T + 47T^{2}
53 1−0.0724iT−53T2 1 - 0.0724iT - 53T^{2}
59 1+6.98T+59T2 1 + 6.98T + 59T^{2}
61 1−8.08iT−61T2 1 - 8.08iT - 61T^{2}
67 1−5.09T+67T2 1 - 5.09T + 67T^{2}
71 1−4.76iT−71T2 1 - 4.76iT - 71T^{2}
73 1+5.59iT−73T2 1 + 5.59iT - 73T^{2}
79 1−17.0T+79T2 1 - 17.0T + 79T^{2}
83 1−10.1T+83T2 1 - 10.1T + 83T^{2}
89 1−11.5T+89T2 1 - 11.5T + 89T^{2}
97 1−0.688iT−97T2 1 - 0.688iT - 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

−9.582595764841833931501094585746, −9.011053814124749980191044687375, −7.900261772619804341337886571194, −7.28051053922899935560748371366, −6.39267926055219585898076523317, −4.76010202162909877495256207836, −4.09369647882330595545681902357, −3.45439140550879819626301373808, −2.32605997206898775737152561212, −0.980921288260781301466566197420, 0.55936531763959491094071632986, 2.87855309289083544190784370148, 3.80169900344041070148985086700, 4.84315641366972731604292843634, 5.64457648830643036425349376821, 6.53881423522370949017263308870, 7.58630422203210480327402793900, 7.84322991035168180047030148981, 8.401812450673517555999448415969, 9.410926946753345201984868739879

Graph of the ZZ-function along the critical line