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Properties

Label 2-3762-33.32-c1-0-27
Degree 22
Conductor 37623762
Sign 0.975+0.218i0.975 + 0.218i
Analytic cond. 30.039730.0397
Root an. cond. 5.480855.48085
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  − 2-s + 4-s + 3.66i·5-s − 2.53i·7-s − 8-s − 3.66i·10-s + (−3.06 + 1.27i)11-s − 4.46i·13-s + 2.53i·14-s + 16-s − 4.49·17-s + i·19-s + 3.66i·20-s + (3.06 − 1.27i)22-s + 1.13i·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.63i·5-s − 0.956i·7-s − 0.353·8-s − 1.15i·10-s + (−0.922 + 0.385i)11-s − 1.23i·13-s + 0.676i·14-s + 0.250·16-s − 1.08·17-s + 0.229i·19-s + 0.819i·20-s + (0.652 − 0.272i)22-s + 0.237i·23-s + ⋯

Functional equation

Λ(s)=(3762s/2 ΓC(s) L(s)=((0.975+0.218i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3762s/2 ΓC(s+1/2) L(s)=((0.975+0.218i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37623762    =    2⋅32⋅11⋅192 \cdot 3^{2} \cdot 11 \cdot 19
Sign: 0.975+0.218i0.975 + 0.218i
Analytic conductor: 30.039730.0397
Root analytic conductor: 5.480855.48085
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3762(989,⋅)\chi_{3762} (989, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3762, ( :1/2), 0.975+0.218i)(2,\ 3762,\ (\ :1/2),\ 0.975 + 0.218i)

Particular Values

L(1)L(1) ≈\approx 0.93206469920.9320646992
L(12)L(\frac12) ≈\approx 0.93206469920.9320646992
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+T 1 + T
3 1 1
11 1+(3.06−1.27i)T 1 + (3.06 - 1.27i)T
19 1−iT 1 - iT
good5 1−3.66iT−5T2 1 - 3.66iT - 5T^{2}
7 1+2.53iT−7T2 1 + 2.53iT - 7T^{2}
13 1+4.46iT−13T2 1 + 4.46iT - 13T^{2}
17 1+4.49T+17T2 1 + 4.49T + 17T^{2}
23 1−1.13iT−23T2 1 - 1.13iT - 23T^{2}
29 1−0.163T+29T2 1 - 0.163T + 29T^{2}
31 1+1.06T+31T2 1 + 1.06T + 31T^{2}
37 1−9.12T+37T2 1 - 9.12T + 37T^{2}
41 1−2.31T+41T2 1 - 2.31T + 41T^{2}
43 1−2.90iT−43T2 1 - 2.90iT - 43T^{2}
47 1−1.49iT−47T2 1 - 1.49iT - 47T^{2}
53 1+2.25iT−53T2 1 + 2.25iT - 53T^{2}
59 1+11.6iT−59T2 1 + 11.6iT - 59T^{2}
61 1+1.04iT−61T2 1 + 1.04iT - 61T^{2}
67 1−3.11T+67T2 1 - 3.11T + 67T^{2}
71 1+8.92iT−71T2 1 + 8.92iT - 71T^{2}
73 1−4.92iT−73T2 1 - 4.92iT - 73T^{2}
79 1−3.42iT−79T2 1 - 3.42iT - 79T^{2}
83 1−8.45T+83T2 1 - 8.45T + 83T^{2}
89 1+15.1iT−89T2 1 + 15.1iT - 89T^{2}
97 1−0.446T+97T2 1 - 0.446T + 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

−8.184586235722084199802905555849, −7.70257436419051498811015732904, −7.17986296151329307122152544221, −6.49118463059028356618715709980, −5.77714825911738872937368430421, −4.62789861133655450006098002216, −3.55726452205827922447116235399, −2.84491782738761230640128297553, −2.07935569643157136470923766197, −0.48390853253535143422705690464, 0.75455244614972764315745600954, 1.94927227210670607210340320560, 2.60976021105525618929461965427, 4.11789075193056093246131454537, 4.79491110818833358494992949056, 5.58172490338217774970303930336, 6.23891761444405934642596415698, 7.23290174915059599509545261257, 8.125465545961774192217562736646, 8.661508913746056431830344755084

Graph of the ZZ-function along the critical line