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Properties

Label 2-2600-13.8-c0-0-1
Degree 22
Conductor 26002600
Sign 0.471+0.881i0.471 + 0.881i
Analytic cond. 1.297561.29756
Root an. cond. 1.139101.13910
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  − 3-s + (1 − i)7-s + (−1 + i)11-s + 13-s − 2i·17-s + (−1 + i)21-s + i·23-s + 27-s + 29-s + (−1 − i)31-s + (1 − i)33-s − 39-s − i·43-s − i·49-s + 2i·51-s + ⋯
L(s)  = 1  − 3-s + (1 − i)7-s + (−1 + i)11-s + 13-s − 2i·17-s + (−1 + i)21-s + i·23-s + 27-s + 29-s + (−1 − i)31-s + (1 − i)33-s − 39-s − i·43-s − i·49-s + 2i·51-s + ⋯

Functional equation

Λ(s)=(2600s/2 ΓC(s) L(s)=((0.471+0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2600s/2 ΓC(s) L(s)=((0.471+0.881i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: 0.471+0.881i0.471 + 0.881i
Analytic conductor: 1.297561.29756
Root analytic conductor: 1.139101.13910
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2600(801,⋅)\chi_{2600} (801, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2600, ( :0), 0.471+0.881i)(2,\ 2600,\ (\ :0),\ 0.471 + 0.881i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83483780010.8348378001
L(12)L(\frac12) ≈\approx 0.83483780010.8348378001
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 1
13 1−T 1 - T
good3 1+T+T2 1 + T + T^{2}
7 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
11 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
17 1+2iT−T2 1 + 2iT - T^{2}
19 1+iT2 1 + iT^{2}
23 1−iT−T2 1 - iT - T^{2}
29 1−T+T2 1 - T + T^{2}
31 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
37 1−iT2 1 - iT^{2}
41 1+iT2 1 + iT^{2}
43 1+iT−T2 1 + iT - T^{2}
47 1−iT2 1 - iT^{2}
53 1+T+T2 1 + T + T^{2}
59 1−iT2 1 - iT^{2}
61 1−T+T2 1 - T + T^{2}
67 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
71 1+iT2 1 + iT^{2}
73 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
79 1−T+T2 1 - T + T^{2}
83 1+iT2 1 + iT^{2}
89 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
97 1+iT2 1 + 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

−8.974335055000032298610816652900, −7.88067564767536930508358346433, −7.45443314238938240696986919500, −6.70372248788775191879185165806, −5.65510493486234228790425747035, −4.99559791657977501237391734408, −4.50403868623672953307677643369, −3.28875099250166150528665345959, −1.99770493597723455456818382992, −0.69893219506802027098242103427, 1.26884347162068186505587203087, 2.47430384534340630315478143657, 3.57220742515065482532847037175, 4.75783534139078831867534267904, 5.41791615040720902154346114464, 6.03051350823542971735633980737, 6.50768722269845412283754053266, 8.004314595349097535579198092902, 8.421238248567540161118263750515, 8.828088370012134247305171807952

Graph of the ZZ-function along the critical line