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Properties

Label 2-3328-8.5-c1-0-10
Degree 22
Conductor 33283328
Sign −0.707−0.707i-0.707 - 0.707i
Analytic cond. 26.574226.5742
Root an. cond. 5.155015.15501
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  + 2.23i·3-s − 3i·5-s − 2.23·7-s − 2.00·9-s + 4.47i·11-s − i·13-s + 6.70·15-s − 3·17-s − 4.47i·19-s − 5.00i·21-s + 8.94·23-s − 4·25-s + 2.23i·27-s + 10i·29-s − 10.0·33-s + ⋯
L(s)  = 1  + 1.29i·3-s − 1.34i·5-s − 0.845·7-s − 0.666·9-s + 1.34i·11-s − 0.277i·13-s + 1.73·15-s − 0.727·17-s − 1.02i·19-s − 1.09i·21-s + 1.86·23-s − 0.800·25-s + 0.430i·27-s + 1.85i·29-s − 1.74·33-s + ⋯

Functional equation

Λ(s)=(3328s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3328s/2 ΓC(s+1/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33283328    =    28⋅132^{8} \cdot 13
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 26.574226.5742
Root analytic conductor: 5.155015.15501
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3328(1665,⋅)\chi_{3328} (1665, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3328, ( :1/2), −0.707−0.707i)(2,\ 3328,\ (\ :1/2),\ -0.707 - 0.707i)

Particular Values

L(1)L(1) ≈\approx 1.0470233991.047023399
L(12)L(\frac12) ≈\approx 1.0470233991.047023399
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
13 1+iT 1 + iT
good3 1−2.23iT−3T2 1 - 2.23iT - 3T^{2}
5 1+3iT−5T2 1 + 3iT - 5T^{2}
7 1+2.23T+7T2 1 + 2.23T + 7T^{2}
11 1−4.47iT−11T2 1 - 4.47iT - 11T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 1+4.47iT−19T2 1 + 4.47iT - 19T^{2}
23 1−8.94T+23T2 1 - 8.94T + 23T^{2}
29 1−10iT−29T2 1 - 10iT - 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+3iT−37T2 1 + 3iT - 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+6.70iT−43T2 1 + 6.70iT - 43T^{2}
47 1−2.23T+47T2 1 - 2.23T + 47T^{2}
53 1+4iT−53T2 1 + 4iT - 53T^{2}
59 1−4.47iT−59T2 1 - 4.47iT - 59T^{2}
61 1−61T2 1 - 61T^{2}
67 1−13.4iT−67T2 1 - 13.4iT - 67T^{2}
71 1+6.70T+71T2 1 + 6.70T + 71T^{2}
73 1+14T+73T2 1 + 14T + 73T^{2}
79 1+8.94T+79T2 1 + 8.94T + 79T^{2}
83 1−17.8iT−83T2 1 - 17.8iT - 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+2T+97T2 1 + 2T + 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

−9.053925337611489955523338613967, −8.669344238771729829735942059260, −7.23020843772506476284103900136, −6.84866079041310868796611732616, −5.45387803173768776976184704064, −4.93670705825449601930891939898, −4.47870854996578677207664046055, −3.59811192331837045214888222113, −2.60814400776212100639895005951, −1.16725832193935022155067138059, 0.34352469053775353781739827941, 1.65138137018106984412920096306, 2.85817313257320118487583343621, 3.13000360775533887724957644628, 4.33468644254046648056491850628, 5.86379107399020928099026297478, 6.26603900581626992646417359105, 6.77980492793573926877724113882, 7.45564683007980874631160375630, 8.137529171714767835562183514671

Graph of the ZZ-function along the critical line