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Properties

Label 2-3168-44.43-c1-0-53
Degree 22
Conductor 31683168
Sign 0.963+0.269i0.963 + 0.269i
Analytic cond. 25.296625.2966
Root an. cond. 5.029575.02957
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  + 3.91·5-s + 3.25·7-s + (1.62 + 2.89i)11-s − 0.831i·13-s − 6.58i·17-s − 1.44·19-s − 0.691i·23-s + 10.3·25-s − 6.15i·29-s − 6.70i·31-s + 12.7·35-s − 6.70·37-s + 3.32i·41-s + 9.62·43-s − 2.73i·47-s + ⋯
L(s)  = 1  + 1.74·5-s + 1.23·7-s + (0.490 + 0.871i)11-s − 0.230i·13-s − 1.59i·17-s − 0.331·19-s − 0.144i·23-s + 2.06·25-s − 1.14i·29-s − 1.20i·31-s + 2.15·35-s − 1.10·37-s + 0.519i·41-s + 1.46·43-s − 0.399i·47-s + ⋯

Functional equation

Λ(s)=(3168s/2 ΓC(s) L(s)=((0.963+0.269i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3168s/2 ΓC(s+1/2) L(s)=((0.963+0.269i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31683168    =    25⋅32⋅112^{5} \cdot 3^{2} \cdot 11
Sign: 0.963+0.269i0.963 + 0.269i
Analytic conductor: 25.296625.2966
Root analytic conductor: 5.029575.02957
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3168(703,⋅)\chi_{3168} (703, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3168, ( :1/2), 0.963+0.269i)(2,\ 3168,\ (\ :1/2),\ 0.963 + 0.269i)

Particular Values

L(1)L(1) ≈\approx 3.2159476713.215947671
L(12)L(\frac12) ≈\approx 3.2159476713.215947671
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 1
11 1+(−1.62−2.89i)T 1 + (-1.62 - 2.89i)T
good5 1−3.91T+5T2 1 - 3.91T + 5T^{2}
7 1−3.25T+7T2 1 - 3.25T + 7T^{2}
13 1+0.831iT−13T2 1 + 0.831iT - 13T^{2}
17 1+6.58iT−17T2 1 + 6.58iT - 17T^{2}
19 1+1.44T+19T2 1 + 1.44T + 19T^{2}
23 1+0.691iT−23T2 1 + 0.691iT - 23T^{2}
29 1+6.15iT−29T2 1 + 6.15iT - 29T^{2}
31 1+6.70iT−31T2 1 + 6.70iT - 31T^{2}
37 1+6.70T+37T2 1 + 6.70T + 37T^{2}
41 1−3.32iT−41T2 1 - 3.32iT - 41T^{2}
43 1−9.62T+43T2 1 - 9.62T + 43T^{2}
47 1+2.73iT−47T2 1 + 2.73iT - 47T^{2}
53 1+1.56T+53T2 1 + 1.56T + 53T^{2}
59 1−3.42iT−59T2 1 - 3.42iT - 59T^{2}
61 1+11.8iT−61T2 1 + 11.8iT - 61T^{2}
67 1−11.9iT−67T2 1 - 11.9iT - 67T^{2}
71 1−10.5iT−71T2 1 - 10.5iT - 71T^{2}
73 1+3.62iT−73T2 1 + 3.62iT - 73T^{2}
79 1−4.91T+79T2 1 - 4.91T + 79T^{2}
83 1+10.3T+83T2 1 + 10.3T + 83T^{2}
89 1+12.9T+89T2 1 + 12.9T + 89T^{2}
97 1−4.60T+97T2 1 - 4.60T + 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

−8.798530902840110518486037243649, −7.86750387482738919025157092066, −7.11048716010049902390093093883, −6.33359552653525661334760088269, −5.50522099737671647616209824692, −4.94845803767096239306602006591, −4.17112864984822609948844810160, −2.57677632317893657515815389498, −2.08972092321898283649317041324, −1.09457972611112948390443081280, 1.46143660557126922811598234409, 1.72759094300985484192400350724, 2.96749201552675547429038775703, 4.08444210041754274359545067154, 5.07836673301544125817236853328, 5.68295688427955764507017696830, 6.30145011778615249913629619271, 7.06627447628439570019050884829, 8.212452091083253534545902207111, 8.775578617728810185180027253396

Graph of the ZZ-function along the critical line