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Properties

Label 2-799-799.798-c0-0-5
Degree 22
Conductor 799799
Sign 11
Analytic cond. 0.3987520.398752
Root an. cond. 0.6314680.631468
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 1.00·4-s + 1.84·5-s + 9-s − 2.61·10-s + 0.765·11-s − 0.999·16-s − 17-s − 1.41·18-s + 1.84·20-s − 1.08·22-s − 1.84·23-s + 2.41·25-s − 0.765·29-s − 0.765·31-s + 1.41·32-s + 1.41·34-s + 1.00·36-s − 1.84·41-s + 0.765·44-s + 1.84·45-s + 2.61·46-s − 47-s + 49-s − 3.41·50-s + 1.41·55-s + 1.08·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.00·4-s + 1.84·5-s + 9-s − 2.61·10-s + 0.765·11-s − 0.999·16-s − 17-s − 1.41·18-s + 1.84·20-s − 1.08·22-s − 1.84·23-s + 2.41·25-s − 0.765·29-s − 0.765·31-s + 1.41·32-s + 1.41·34-s + 1.00·36-s − 1.84·41-s + 0.765·44-s + 1.84·45-s + 2.61·46-s − 47-s + 49-s − 3.41·50-s + 1.41·55-s + 1.08·58-s + ⋯

Functional equation

Λ(s)=(799s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(799s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 799799    =    17⋅4717 \cdot 47
Sign: 11
Analytic conductor: 0.3987520.398752
Root analytic conductor: 0.6314680.631468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ799(798,⋅)\chi_{799} (798, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 799, ( :0), 1)(2,\ 799,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.70195316810.7019531681
L(12)L(\frac12) ≈\approx 0.70195316810.7019531681
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)
bad17 1+T 1 + T
47 1+T 1 + T
good2 1+1.41T+T2 1 + 1.41T + T^{2}
3 1−T2 1 - T^{2}
5 1−1.84T+T2 1 - 1.84T + T^{2}
7 1−T2 1 - T^{2}
11 1−0.765T+T2 1 - 0.765T + T^{2}
13 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+1.84T+T2 1 + 1.84T + T^{2}
29 1+0.765T+T2 1 + 0.765T + T^{2}
31 1+0.765T+T2 1 + 0.765T + T^{2}
37 1−T2 1 - T^{2}
41 1+1.84T+T2 1 + 1.84T + T^{2}
43 1−T2 1 - T^{2}
53 1+T2 1 + T^{2}
59 1−1.41T+T2 1 - 1.41T + T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−0.765T+T2 1 - 0.765T + T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - T^{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

−10.14064795174349330118062943248, −9.612807905988525046426979837388, −9.078286563532982412464251826877, −8.185903378605004678847052633019, −6.95202882038181500529883090318, −6.49664325122377342082172518890, −5.36033789114543979090999479686, −4.09745807698964468747766934923, −2.11678113221751284338357849876, −1.58425295725740731501318166152, 1.58425295725740731501318166152, 2.11678113221751284338357849876, 4.09745807698964468747766934923, 5.36033789114543979090999479686, 6.49664325122377342082172518890, 6.95202882038181500529883090318, 8.185903378605004678847052633019, 9.078286563532982412464251826877, 9.612807905988525046426979837388, 10.14064795174349330118062943248

Graph of the ZZ-function along the critical line