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Properties

Label 2-1127-161.160-c1-0-62
Degree 22
Conductor 11271127
Sign 0.912+0.409i0.912 + 0.409i
Analytic cond. 8.999148.99914
Root an. cond. 2.999852.99985
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  + 2.54·2-s − 0.893i·3-s + 4.49·4-s − 2.27i·6-s + 6.36·8-s + 2.20·9-s − 4.02i·12-s + 2.48i·13-s + 7.23·16-s + 5.61·18-s + 4.79·23-s − 5.69i·24-s − 5·25-s + 6.32i·26-s − 4.64i·27-s + ⋯
L(s)  = 1  + 1.80·2-s − 0.516i·3-s + 2.24·4-s − 0.930i·6-s + 2.25·8-s + 0.733·9-s − 1.16i·12-s + 0.688i·13-s + 1.80·16-s + 1.32·18-s + 1.00·23-s − 1.16i·24-s − 25-s + 1.24i·26-s − 0.894i·27-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=((0.912+0.409i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s+1/2) L(s)=((0.912+0.409i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: 0.912+0.409i0.912 + 0.409i
Analytic conductor: 8.999148.99914
Root analytic conductor: 2.999852.99985
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1127(1126,⋅)\chi_{1127} (1126, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1127, ( :1/2), 0.912+0.409i)(2,\ 1127,\ (\ :1/2),\ 0.912 + 0.409i)

Particular Values

L(1)L(1) ≈\approx 5.0120899215.012089921
L(12)L(\frac12) ≈\approx 5.0120899215.012089921
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)
bad7 1 1
23 1−4.79T 1 - 4.79T
good2 1−2.54T+2T2 1 - 2.54T + 2T^{2}
3 1+0.893iT−3T2 1 + 0.893iT - 3T^{2}
5 1+5T2 1 + 5T^{2}
11 1−11T2 1 - 11T^{2}
13 1−2.48iT−13T2 1 - 2.48iT - 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+19T2 1 + 19T^{2}
29 1+6.70T+29T2 1 + 6.70T + 29T^{2}
31 1+4.54iT−31T2 1 + 4.54iT - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1+12.7iT−41T2 1 + 12.7iT - 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1−10.6iT−47T2 1 - 10.6iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1−14.7iT−59T2 1 - 14.7iT - 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+14.0T+71T2 1 + 14.0T + 71T^{2}
73 1−17.0iT−73T2 1 - 17.0iT - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+97T2 1 + 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

−9.973345816302181882365291165928, −8.934562156413334514007846932655, −7.45871460866168554735379852737, −7.19962773347397530538165154963, −6.17894013779057622726054391222, −5.51345892862312186846599405438, −4.40015312751536009286051866415, −3.85615029850207714527325454267, −2.58501813721742578294643022206, −1.62042927261895701082694012344, 1.75897450465415513833261066341, 3.10811893878716614086536469966, 3.77014047699622932868807005737, 4.73780615004010974241403511480, 5.30151817176759316305364423517, 6.25972177178299932695338590010, 7.09711631998868157309926536954, 7.898089952528120247314598844739, 9.243693549247054813164598737039, 10.16318742648892212340432778693

Graph of the ZZ-function along the critical line