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Properties

Label 2-435-1.1-c3-0-46
Degree 22
Conductor 435435
Sign −1-1
Analytic cond. 25.665825.6658
Root an. cond. 5.066145.06614
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s − 7·4-s + 5·5-s − 3·6-s + 4·7-s + 15·8-s + 9·9-s − 5·10-s − 36·11-s − 21·12-s − 22·13-s − 4·14-s + 15·15-s + 41·16-s − 2·17-s − 9·18-s − 56·19-s − 35·20-s + 12·21-s + 36·22-s − 40·23-s + 45·24-s + 25·25-s + 22·26-s + 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.447·5-s − 0.204·6-s + 0.215·7-s + 0.662·8-s + 1/3·9-s − 0.158·10-s − 0.986·11-s − 0.505·12-s − 0.469·13-s − 0.0763·14-s + 0.258·15-s + 0.640·16-s − 0.0285·17-s − 0.117·18-s − 0.676·19-s − 0.391·20-s + 0.124·21-s + 0.348·22-s − 0.362·23-s + 0.382·24-s + 1/5·25-s + 0.165·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 25.665825.6658
Root analytic conductor: 5.066145.06614
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 435, ( :3/2), −1)(2,\ 435,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1−pT 1 - p T
5 1−pT 1 - p T
29 1−pT 1 - p T
good2 1+T+p3T2 1 + T + p^{3} T^{2}
7 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
11 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
13 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
17 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
19 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
23 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
31 1−152T+p3T2 1 - 152 T + p^{3} T^{2}
37 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
41 1+250T+p3T2 1 + 250 T + p^{3} T^{2}
43 1+412T+p3T2 1 + 412 T + p^{3} T^{2}
47 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
53 1+762T+p3T2 1 + 762 T + p^{3} T^{2}
59 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
61 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
67 1+244T+p3T2 1 + 244 T + p^{3} T^{2}
71 1−600T+p3T2 1 - 600 T + p^{3} T^{2}
73 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
79 1+640T+p3T2 1 + 640 T + p^{3} T^{2}
83 1−8pT+p3T2 1 - 8 p T + p^{3} T^{2}
89 1−150T+p3T2 1 - 150 T + p^{3} T^{2}
97 1+1690T+p3T2 1 + 1690 T + p^{3} 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

−10.03337444848525061151970572573, −9.453757669856857417113204759096, −8.323091959849150548058122240642, −7.959638156522876811417947654592, −6.62038039851849601214079969333, −5.24153362785041470946393272646, −4.47289470116913976637026712426, −3.05582687556138009517029285414, −1.71359044527595527357752911774, 0, 1.71359044527595527357752911774, 3.05582687556138009517029285414, 4.47289470116913976637026712426, 5.24153362785041470946393272646, 6.62038039851849601214079969333, 7.959638156522876811417947654592, 8.323091959849150548058122240642, 9.453757669856857417113204759096, 10.03337444848525061151970572573

Graph of the ZZ-function along the critical line