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Properties

Label 2-528-44.43-c1-0-3
Degree 22
Conductor 528528
Sign −0.0791−0.996i-0.0791 - 0.996i
Analytic cond. 4.216104.21610
Root an. cond. 2.053312.05331
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  + i·3-s + 2.73·5-s − 5.13·7-s − 9-s + (1.88 + 2.73i)11-s + 5.13i·13-s + 2.73i·15-s + 3.76i·17-s + 3.76·19-s − 5.13i·21-s + 1.26i·23-s + 2.46·25-s − i·27-s + 2i·31-s + (−2.73 + 1.88i)33-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.22·5-s − 1.94·7-s − 0.333·9-s + (0.566 + 0.823i)11-s + 1.42i·13-s + 0.705i·15-s + 0.912i·17-s + 0.862·19-s − 1.12i·21-s + 0.264i·23-s + 0.492·25-s − 0.192i·27-s + 0.359i·31-s + (−0.475 + 0.327i)33-s + ⋯

Functional equation

Λ(s)=(528s/2 ΓC(s) L(s)=((−0.0791−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(528s/2 ΓC(s+1/2) L(s)=((−0.0791−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 528528    =    24⋅3⋅112^{4} \cdot 3 \cdot 11
Sign: −0.0791−0.996i-0.0791 - 0.996i
Analytic conductor: 4.216104.21610
Root analytic conductor: 2.053312.05331
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ528(175,⋅)\chi_{528} (175, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 528, ( :1/2), −0.0791−0.996i)(2,\ 528,\ (\ :1/2),\ -0.0791 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 0.903446+0.978003i0.903446 + 0.978003i
L(12)L(\frac12) ≈\approx 0.903446+0.978003i0.903446 + 0.978003i
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−iT 1 - iT
11 1+(−1.88−2.73i)T 1 + (-1.88 - 2.73i)T
good5 1−2.73T+5T2 1 - 2.73T + 5T^{2}
7 1+5.13T+7T2 1 + 5.13T + 7T^{2}
13 1−5.13iT−13T2 1 - 5.13iT - 13T^{2}
17 1−3.76iT−17T2 1 - 3.76iT - 17T^{2}
19 1−3.76T+19T2 1 - 3.76T + 19T^{2}
23 1−1.26iT−23T2 1 - 1.26iT - 23T^{2}
29 1−29T2 1 - 29T^{2}
31 1−2iT−31T2 1 - 2iT - 31T^{2}
37 1−0.535T+37T2 1 - 0.535T + 37T^{2}
41 1+10.2iT−41T2 1 + 10.2iT - 41T^{2}
43 1−3.76T+43T2 1 - 3.76T + 43T^{2}
47 1−4.19iT−47T2 1 - 4.19iT - 47T^{2}
53 1+10.7T+53T2 1 + 10.7T + 53T^{2}
59 1−9.46iT−59T2 1 - 9.46iT - 59T^{2}
61 1+2.38iT−61T2 1 + 2.38iT - 61T^{2}
67 1+10.3iT−67T2 1 + 10.3iT - 67T^{2}
71 1+10.7iT−71T2 1 + 10.7iT - 71T^{2}
73 1−10.2iT−73T2 1 - 10.2iT - 73T^{2}
79 1−5.13T+79T2 1 - 5.13T + 79T^{2}
83 1−10.2T+83T2 1 - 10.2T + 83T^{2}
89 1+10.3T+89T2 1 + 10.3T + 89T^{2}
97 1−5.46T+97T2 1 - 5.46T + 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

−10.80500661928748324966273187158, −9.861901780650944139559865929486, −9.507081394465165023006283540260, −8.973703217072276653309184726314, −7.15569410309150290903651844771, −6.39365909006246876636569323092, −5.72827483457004752827191862035, −4.31501378893847114808764717566, −3.29151701807960160693323846979, −1.92986857107540914392678257284, 0.78097426748412010368823622902, 2.68928646499871485696280740935, 3.36546361051797070805971778504, 5.39255675635810422968123581218, 6.11176204638169074275295459624, 6.69204670312245858379595777662, 7.85385292366142228116700352198, 9.142906844389981402838589704953, 9.657246507533088444846044839782, 10.38143953300174778915351796238

Graph of the ZZ-function along the critical line