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Properties

Label 2-286650-1.1-c1-0-127
Degree 22
Conductor 286650286650
Sign 11
Analytic cond. 2288.912288.91
Root an. cond. 47.842547.8425
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 2·11-s + 13-s + 16-s − 5·17-s + 5·19-s − 2·22-s − 4·23-s + 26-s + 2·29-s + 4·31-s + 32-s − 5·34-s + 4·37-s + 5·38-s + 3·41-s + 11·43-s − 2·44-s − 4·46-s + 2·47-s + 52-s − 6·53-s + 2·58-s + 6·59-s + 4·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 0.277·13-s + 1/4·16-s − 1.21·17-s + 1.14·19-s − 0.426·22-s − 0.834·23-s + 0.196·26-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.857·34-s + 0.657·37-s + 0.811·38-s + 0.468·41-s + 1.67·43-s − 0.301·44-s − 0.589·46-s + 0.291·47-s + 0.138·52-s − 0.824·53-s + 0.262·58-s + 0.781·59-s + 0.508·62-s + ⋯

Functional equation

Λ(s)=(286650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(286650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 286650286650    =    2⋅32⋅52⋅72⋅132 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13
Sign: 11
Analytic conductor: 2288.912288.91
Root analytic conductor: 47.842547.8425
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 286650, ( :1/2), 1)(2,\ 286650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.3849262514.384926251
L(12)L(\frac12) ≈\approx 4.3849262514.384926251
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−T 1 - T
3 1 1
5 1 1
7 1 1
13 1−T 1 - T
good11 1+2T+pT2 1 + 2 T + p T^{2}
17 1+5T+pT2 1 + 5 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−11T+pT2 1 - 11 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1−15T+pT2 1 - 15 T + p T^{2}
79 1+11T+pT2 1 + 11 T + p T^{2}
83 1−3T+pT2 1 - 3 T + p T^{2}
89 1+T+pT2 1 + T + p T^{2}
97 1+13T+pT2 1 + 13 T + p 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

−12.67917813932837, −12.43582046243695, −11.82746747041095, −11.34172966018200, −11.00693333208005, −10.62888577606515, −9.910686626583208, −9.630931870993208, −9.089434731768212, −8.426972184690198, −7.995887465229388, −7.632534040468041, −6.989647531061121, −6.546589940136528, −6.109550543311592, −5.515936960223627, −5.155290801940360, −4.557033622062694, −4.024981177447809, −3.702249931590108, −2.735099587140164, −2.628910648735625, −1.948995965877139, −1.131184642810546, −0.5206274634854719, 0.5206274634854719, 1.131184642810546, 1.948995965877139, 2.628910648735625, 2.735099587140164, 3.702249931590108, 4.024981177447809, 4.557033622062694, 5.155290801940360, 5.515936960223627, 6.109550543311592, 6.546589940136528, 6.989647531061121, 7.632534040468041, 7.995887465229388, 8.426972184690198, 9.089434731768212, 9.630931870993208, 9.910686626583208, 10.62888577606515, 11.00693333208005, 11.34172966018200, 11.82746747041095, 12.43582046243695, 12.67917813932837

Graph of the ZZ-function along the critical line