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Properties

Label 2-3200-8.5-c1-0-73
Degree 22
Conductor 32003200
Sign −0.707−0.707i-0.707 - 0.707i
Analytic cond. 25.552125.5521
Root an. cond. 5.054915.05491
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  − 1.41i·3-s − 4.24·7-s + 0.999·9-s − 5.65i·11-s − 2i·13-s − 6·17-s − 2.82i·19-s + 6i·21-s + 7.07·23-s − 5.65i·27-s + 4i·29-s + 2.82·31-s − 8.00·33-s − 2i·37-s − 2.82·39-s + ⋯
L(s)  = 1  − 0.816i·3-s − 1.60·7-s + 0.333·9-s − 1.70i·11-s − 0.554i·13-s − 1.45·17-s − 0.648i·19-s + 1.30i·21-s + 1.47·23-s − 1.08i·27-s + 0.742i·29-s + 0.508·31-s − 1.39·33-s − 0.328i·37-s − 0.452·39-s + ⋯

Functional equation

Λ(s)=(3200s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3200s/2 ΓC(s+1/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{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: 32003200    =    27⋅522^{7} \cdot 5^{2}
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 25.552125.5521
Root analytic conductor: 5.054915.05491
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3200(1601,⋅)\chi_{3200} (1601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3200, ( :1/2), −0.707−0.707i)(2,\ 3200,\ (\ :1/2),\ -0.707 - 0.707i)

Particular Values

L(1)L(1) ≈\approx 0.48693168890.4869316889
L(12)L(\frac12) ≈\approx 0.48693168890.4869316889
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
5 1 1
good3 1+1.41iT−3T2 1 + 1.41iT - 3T^{2}
7 1+4.24T+7T2 1 + 4.24T + 7T^{2}
11 1+5.65iT−11T2 1 + 5.65iT - 11T^{2}
13 1+2iT−13T2 1 + 2iT - 13T^{2}
17 1+6T+17T2 1 + 6T + 17T^{2}
19 1+2.82iT−19T2 1 + 2.82iT - 19T^{2}
23 1−7.07T+23T2 1 - 7.07T + 23T^{2}
29 1−4iT−29T2 1 - 4iT - 29T^{2}
31 1−2.82T+31T2 1 - 2.82T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+8T+41T2 1 + 8T + 41T^{2}
43 1+1.41iT−43T2 1 + 1.41iT - 43T^{2}
47 1+1.41T+47T2 1 + 1.41T + 47T^{2}
53 1−2iT−53T2 1 - 2iT - 53T^{2}
59 1+2.82iT−59T2 1 + 2.82iT - 59T^{2}
61 1−14iT−61T2 1 - 14iT - 61T^{2}
67 1−4.24iT−67T2 1 - 4.24iT - 67T^{2}
71 1+2.82T+71T2 1 + 2.82T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1+16.9T+79T2 1 + 16.9T + 79T^{2}
83 1−12.7iT−83T2 1 - 12.7iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 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

−8.357979040210222510042235318109, −7.06595783243019650090753817116, −6.85587097202252289816848869102, −6.13750876012166040545444477804, −5.34875685075280458305508789334, −4.15549126709171501446856606202, −3.14214245085775495247024182010, −2.66687396725846354100161100346, −1.11046800699204804839025176366, −0.16232167247119800372126578819, 1.75155587832882000337387975984, 2.82336084462008636173033952304, 3.77263282140169461112918312702, 4.46536479420125400518651568974, 5.05221487771172827511651040678, 6.38784479031823761813414918640, 6.77468514865452467429312961780, 7.41834551537185633111699430467, 8.682859984679428020459946166776, 9.352974750745270284735656756354

Graph of the ZZ-function along the critical line