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Properties

Label 2-40e2-5.4-c1-0-27
Degree 22
Conductor 16001600
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·3-s + 2i·7-s − 9-s − 4·11-s − 6i·13-s + 2i·17-s − 8·19-s − 4·21-s − 6i·23-s + 4i·27-s − 2·29-s − 4·31-s − 8i·33-s − 2i·37-s + 12·39-s + ⋯
L(s)  = 1  + 1.15i·3-s + 0.755i·7-s − 0.333·9-s − 1.20·11-s − 1.66i·13-s + 0.485i·17-s − 1.83·19-s − 0.872·21-s − 1.25i·23-s + 0.769i·27-s − 0.371·29-s − 0.718·31-s − 1.39i·33-s − 0.328i·37-s + 1.92·39-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1600(449,⋅)\chi_{1600} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 1600, ( :1/2), −0.447+0.894i)(2,\ 1600,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 1−2iT−3T2 1 - 2iT - 3T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1+6iT−13T2 1 + 6iT - 13T^{2}
17 1−2iT−17T2 1 - 2iT - 17T^{2}
19 1+8T+19T2 1 + 8T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+2iT−37T2 1 + 2iT - 37T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1−2iT−43T2 1 - 2iT - 43T^{2}
47 1−2iT−47T2 1 - 2iT - 47T^{2}
53 1−2iT−53T2 1 - 2iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+6iT−67T2 1 + 6iT - 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1+10iT−73T2 1 + 10iT - 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−10iT−83T2 1 - 10iT - 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1−10iT−97T2 1 - 10iT - 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

−9.155985583096117039394650441908, −8.409799530723711511241889912968, −7.87828404512942150249527675303, −6.56907834446500933513654921958, −5.60104124022750292102211641871, −5.06632952221844786878583681668, −4.13537087465230804571820996375, −3.12607651914747200329058045298, −2.21913351617222597891132330615, 0, 1.59187482730967088947058848206, 2.32443200779169547829008970144, 3.77112147073722392183692579348, 4.65346565249472860870920095398, 5.73752957257773745987183120060, 6.86375252968661253189757882435, 7.03480239937336302191128047163, 7.947159410057509323919136631618, 8.680928383242480127954456467440

Graph of the ZZ-function along the critical line