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Properties

Label 2-1280-40.37-c0-0-1
Degree 22
Conductor 12801280
Sign −0.229+0.973i-0.229 + 0.973i
Analytic cond. 0.6388030.638803
Root an. cond. 0.7992510.799251
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − i·9-s + (−1 − i)13-s + (−1 − i)17-s + 25-s + (1 − i)37-s + i·45-s − i·49-s + (−1 − i)53-s + (1 + i)65-s + (−1 + i)73-s − 81-s + (1 + i)85-s + (1 + i)97-s + 2i·101-s + ⋯
L(s)  = 1  − 5-s − i·9-s + (−1 − i)13-s + (−1 − i)17-s + 25-s + (1 − i)37-s + i·45-s − i·49-s + (−1 − i)53-s + (1 + i)65-s + (−1 + i)73-s − 81-s + (1 + i)85-s + (1 + i)97-s + 2i·101-s + ⋯

Functional equation

Λ(s)=(1280s/2 ΓC(s) L(s)=((−0.229+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1280s/2 ΓC(s) L(s)=((−0.229+0.973i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12801280    =    28⋅52^{8} \cdot 5
Sign: −0.229+0.973i-0.229 + 0.973i
Analytic conductor: 0.6388030.638803
Root analytic conductor: 0.7992510.799251
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1280(897,⋅)\chi_{1280} (897, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1280, ( :0), −0.229+0.973i)(2,\ 1280,\ (\ :0),\ -0.229 + 0.973i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.62855845190.6285584519
L(12)L(\frac12) ≈\approx 0.62855845190.6285584519
L(1)L(1) 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+T 1 + T
good3 1+iT2 1 + iT^{2}
7 1+iT2 1 + iT^{2}
11 1−T2 1 - T^{2}
13 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
17 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
19 1+T2 1 + T^{2}
23 1−iT2 1 - iT^{2}
29 1+T2 1 + T^{2}
31 1+T2 1 + T^{2}
37 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
41 1+T2 1 + T^{2}
43 1+iT2 1 + iT^{2}
47 1+iT2 1 + iT^{2}
53 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
59 1+T2 1 + T^{2}
61 1−T2 1 - T^{2}
67 1−iT2 1 - iT^{2}
71 1+T2 1 + T^{2}
73 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
79 1−T2 1 - T^{2}
83 1+iT2 1 + iT^{2}
89 1−T2 1 - T^{2}
97 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{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

−9.549043106157519284353947974808, −8.862133571389173939098725474424, −7.925364502967568629946506837545, −7.23804426260962121104767848944, −6.49651926228537385469881830752, −5.30401274665158675155824955763, −4.45135557577723848392513528084, −3.48790581430571638952733573985, −2.55676696881410933695317694536, −0.52541391397976017058311200122, 1.86337020966777733282503932303, 2.99792808947332476694819824521, 4.43813509290071215277046417838, 4.58972446652015617176234769491, 6.01829734414518287410648994780, 6.98510380780720997632179167339, 7.67555328746532838095396762153, 8.390080089110390140436706790188, 9.194119156098533152783828734163, 10.17838798312829118638460205802

Graph of the ZZ-function along the critical line