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Properties

Label 2-3328-8.5-c1-0-49
Degree 22
Conductor 33283328
Sign 0.707−0.707i0.707 - 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  + i·3-s + i·5-s + 3·7-s + 2·9-s − 2i·11-s − i·13-s − 15-s − 3·17-s + 2i·19-s + 3i·21-s + 4·23-s + 4·25-s + 5i·27-s − 2i·29-s − 4·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s + 1.13·7-s + 0.666·9-s − 0.603i·11-s − 0.277i·13-s − 0.258·15-s − 0.727·17-s + 0.458i·19-s + 0.654i·21-s + 0.834·23-s + 0.800·25-s + 0.962i·27-s − 0.371i·29-s − 0.718·31-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: 0.707−0.707i0.707 - 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 2.4007898232.400789823
L(12)L(\frac12) ≈\approx 2.4007898232.400789823
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−iT−3T2 1 - iT - 3T^{2}
5 1−iT−5T2 1 - iT - 5T^{2}
7 1−3T+7T2 1 - 3T + 7T^{2}
11 1+2iT−11T2 1 + 2iT - 11T^{2}
17 1+3T+17T2 1 + 3T + 17T^{2}
19 1−2iT−19T2 1 - 2iT - 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
29 1+2iT−29T2 1 + 2iT - 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1−5iT−37T2 1 - 5iT - 37T^{2}
41 1−12T+41T2 1 - 12T + 41T^{2}
43 1+7iT−43T2 1 + 7iT - 43T^{2}
47 1−9T+47T2 1 - 9T + 47T^{2}
53 1−4iT−53T2 1 - 4iT - 53T^{2}
59 1+6iT−59T2 1 + 6iT - 59T^{2}
61 1−4iT−61T2 1 - 4iT - 61T^{2}
67 1+10iT−67T2 1 + 10iT - 67T^{2}
71 1+15T+71T2 1 + 15T + 71T^{2}
73 1−2T+73T2 1 - 2T + 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1+2T+89T2 1 + 2T + 89T^{2}
97 1−10T+97T2 1 - 10T + 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

−8.880157672028371211683869038403, −7.905114463359413762411237583990, −7.35400998102095319157005823962, −6.49476624921280522772804785712, −5.56761019982169990166111603987, −4.80604899610561460037535656113, −4.15457340430832945190843238100, −3.24796225218268772921656850650, −2.19254625144669468143236401663, −1.04747734778876854886060885548, 0.944510207715478198284809024647, 1.76602891595524199945264321345, 2.63266241948440037720055765942, 4.17543957128242856597877921997, 4.60549786053395597259857643766, 5.38607867962064949987032330061, 6.42416732087092013598387706853, 7.32767881993696853970978007488, 7.50211611712680101994387131035, 8.628387802360297444039085334453

Graph of the ZZ-function along the critical line