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Properties

Label 2-700-1.1-c5-0-16
Degree 22
Conductor 700700
Sign 11
Analytic cond. 112.268112.268
Root an. cond. 10.595610.5956
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.1·3-s − 49·7-s + 162.·9-s − 427.·11-s + 646.·13-s + 1.02e3·17-s + 2.25e3·19-s − 986.·21-s − 2.59e3·23-s − 1.62e3·27-s + 869.·29-s + 4.40e3·31-s − 8.59e3·33-s − 2.55e3·37-s + 1.30e4·39-s + 6.22e3·41-s − 7.75e3·43-s − 1.88e3·47-s + 2.40e3·49-s + 2.07e4·51-s + 1.17e4·53-s + 4.53e4·57-s + 3.33e4·59-s − 1.21e3·61-s − 7.93e3·63-s + 5.04e4·67-s − 5.22e4·69-s + ⋯
L(s)  = 1  + 1.29·3-s − 0.377·7-s + 0.666·9-s − 1.06·11-s + 1.06·13-s + 0.863·17-s + 1.43·19-s − 0.487·21-s − 1.02·23-s − 0.430·27-s + 0.192·29-s + 0.823·31-s − 1.37·33-s − 0.306·37-s + 1.37·39-s + 0.578·41-s − 0.639·43-s − 0.124·47-s + 0.142·49-s + 1.11·51-s + 0.573·53-s + 1.84·57-s + 1.24·59-s − 0.0417·61-s − 0.251·63-s + 1.37·67-s − 1.32·69-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 112.268112.268
Root analytic conductor: 10.595610.5956
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 700, ( :5/2), 1)(2,\ 700,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.5470982073.547098207
L(12)L(\frac12) ≈\approx 3.5470982073.547098207
L(72)L(\frac{7}{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
7 1+49T 1 + 49T
good3 1−20.1T+243T2 1 - 20.1T + 243T^{2}
11 1+427.T+1.61e5T2 1 + 427.T + 1.61e5T^{2}
13 1−646.T+3.71e5T2 1 - 646.T + 3.71e5T^{2}
17 1−1.02e3T+1.41e6T2 1 - 1.02e3T + 1.41e6T^{2}
19 1−2.25e3T+2.47e6T2 1 - 2.25e3T + 2.47e6T^{2}
23 1+2.59e3T+6.43e6T2 1 + 2.59e3T + 6.43e6T^{2}
29 1−869.T+2.05e7T2 1 - 869.T + 2.05e7T^{2}
31 1−4.40e3T+2.86e7T2 1 - 4.40e3T + 2.86e7T^{2}
37 1+2.55e3T+6.93e7T2 1 + 2.55e3T + 6.93e7T^{2}
41 1−6.22e3T+1.15e8T2 1 - 6.22e3T + 1.15e8T^{2}
43 1+7.75e3T+1.47e8T2 1 + 7.75e3T + 1.47e8T^{2}
47 1+1.88e3T+2.29e8T2 1 + 1.88e3T + 2.29e8T^{2}
53 1−1.17e4T+4.18e8T2 1 - 1.17e4T + 4.18e8T^{2}
59 1−3.33e4T+7.14e8T2 1 - 3.33e4T + 7.14e8T^{2}
61 1+1.21e3T+8.44e8T2 1 + 1.21e3T + 8.44e8T^{2}
67 1−5.04e4T+1.35e9T2 1 - 5.04e4T + 1.35e9T^{2}
71 1−5.72e4T+1.80e9T2 1 - 5.72e4T + 1.80e9T^{2}
73 1+8.40e4T+2.07e9T2 1 + 8.40e4T + 2.07e9T^{2}
79 1−9.84e4T+3.07e9T2 1 - 9.84e4T + 3.07e9T^{2}
83 1−3.21e4T+3.93e9T2 1 - 3.21e4T + 3.93e9T^{2}
89 1−5.43e4T+5.58e9T2 1 - 5.43e4T + 5.58e9T^{2}
97 1−2.17e4T+8.58e9T2 1 - 2.17e4T + 8.58e9T^{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

−9.715075521473317444322645544191, −8.679816580525358449667233193214, −8.051847013109266841526577002034, −7.39678394169658436991657166323, −6.12034048871045486050128002455, −5.18875134753975272372199552175, −3.74763557682666703380903945841, −3.14576833473761073611223871031, −2.17093829944015525415507595411, −0.834811560736841051126770626879, 0.834811560736841051126770626879, 2.17093829944015525415507595411, 3.14576833473761073611223871031, 3.74763557682666703380903945841, 5.18875134753975272372199552175, 6.12034048871045486050128002455, 7.39678394169658436991657166323, 8.051847013109266841526577002034, 8.679816580525358449667233193214, 9.715075521473317444322645544191

Graph of the ZZ-function along the critical line