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Properties

Label 2-2112-33.32-c1-0-0
Degree 22
Conductor 21122112
Sign −0.120+0.992i-0.120 + 0.992i
Analytic cond. 16.864416.8644
Root an. cond. 4.106624.10662
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  + (−0.393 + 1.68i)3-s + 2.18i·5-s + 1.18i·7-s + (−2.69 − 1.32i)9-s + (−3.29 + 0.359i)11-s + 4.84i·13-s + (−3.69 − 0.860i)15-s − 6.16·17-s − 5.09i·19-s + (−1.99 − 0.466i)21-s − 3.84i·23-s + 0.213·25-s + (3.29 − 4.01i)27-s − 3.57·29-s + 7.38·31-s + ⋯
L(s)  = 1  + (−0.227 + 0.973i)3-s + 0.978i·5-s + 0.448i·7-s + (−0.896 − 0.442i)9-s + (−0.994 + 0.108i)11-s + 1.34i·13-s + (−0.952 − 0.222i)15-s − 1.49·17-s − 1.16i·19-s + (−0.436 − 0.101i)21-s − 0.800i·23-s + 0.0426·25-s + (0.634 − 0.772i)27-s − 0.663·29-s + 1.32·31-s + ⋯

Functional equation

Λ(s)=(2112s/2 ΓC(s) L(s)=((−0.120+0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2112s/2 ΓC(s+1/2) L(s)=((−0.120+0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21122112    =    26⋅3⋅112^{6} \cdot 3 \cdot 11
Sign: −0.120+0.992i-0.120 + 0.992i
Analytic conductor: 16.864416.8644
Root analytic conductor: 4.106624.10662
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2112(65,⋅)\chi_{2112} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2112, ( :1/2), −0.120+0.992i)(2,\ 2112,\ (\ :1/2),\ -0.120 + 0.992i)

Particular Values

L(1)L(1) ≈\approx 0.21939807350.2193980735
L(12)L(\frac12) ≈\approx 0.21939807350.2193980735
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+(0.393−1.68i)T 1 + (0.393 - 1.68i)T
11 1+(3.29−0.359i)T 1 + (3.29 - 0.359i)T
good5 1−2.18iT−5T2 1 - 2.18iT - 5T^{2}
7 1−1.18iT−7T2 1 - 1.18iT - 7T^{2}
13 1−4.84iT−13T2 1 - 4.84iT - 13T^{2}
17 1+6.16T+17T2 1 + 6.16T + 17T^{2}
19 1+5.09iT−19T2 1 + 5.09iT - 19T^{2}
23 1+3.84iT−23T2 1 + 3.84iT - 23T^{2}
29 1+3.57T+29T2 1 + 3.57T + 29T^{2}
31 1−7.38T+31T2 1 - 7.38T + 31T^{2}
37 1+5.38T+37T2 1 + 5.38T + 37T^{2}
41 1+0.426T+41T2 1 + 0.426T + 41T^{2}
43 1+6.02iT−43T2 1 + 6.02iT - 43T^{2}
47 1−7.93iT−47T2 1 - 7.93iT - 47T^{2}
53 1+4.84iT−53T2 1 + 4.84iT - 53T^{2}
59 1+1.00iT−59T2 1 + 1.00iT - 59T^{2}
61 1−8.65iT−61T2 1 - 8.65iT - 61T^{2}
67 1+1.80T+67T2 1 + 1.80T + 67T^{2}
71 1−5.27iT−71T2 1 - 5.27iT - 71T^{2}
73 1−13.4iT−73T2 1 - 13.4iT - 73T^{2}
79 1+12.3iT−79T2 1 + 12.3iT - 79T^{2}
83 1+0.852T+83T2 1 + 0.852T + 83T^{2}
89 1−7.96iT−89T2 1 - 7.96iT - 89T^{2}
97 1+17.5T+97T2 1 + 17.5T + 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

−9.671507956700806263686842050548, −8.937684898960180994249193656797, −8.408220742906205223238037410235, −6.99411761058759900858227996404, −6.67694200431570123159224347769, −5.65664257297466190932762398610, −4.71852818973692271185657173326, −4.14436184515980069001177066909, −2.83311308090044823630461370161, −2.35563902260707713452805335139, 0.082188752155315212592204741764, 1.18030014548915274443944834480, 2.31097336510494083599501361944, 3.41948948968686708448047667192, 4.70464669097435032830620965804, 5.39033730936162944865115917321, 6.08509708949126289565284729636, 7.07761447388565163569102998129, 7.914000920364921147999295637852, 8.261637617923293288910138990740

Graph of the ZZ-function along the critical line