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Properties

Label 2-435-1.1-c3-0-29
Degree 22
Conductor 435435
Sign −1-1
Analytic cond. 25.665825.6658
Root an. cond. 5.066145.06614
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.691·2-s − 3·3-s − 7.52·4-s + 5·5-s + 2.07·6-s − 14.5·7-s + 10.7·8-s + 9·9-s − 3.45·10-s + 36.8·11-s + 22.5·12-s − 23.9·13-s + 10.0·14-s − 15·15-s + 52.7·16-s + 62.8·17-s − 6.22·18-s − 94.1·19-s − 37.6·20-s + 43.7·21-s − 25.4·22-s + 185.·23-s − 32.2·24-s + 25·25-s + 16.5·26-s − 27·27-s + 109.·28-s + ⋯
L(s)  = 1  − 0.244·2-s − 0.577·3-s − 0.940·4-s + 0.447·5-s + 0.141·6-s − 0.786·7-s + 0.474·8-s + 0.333·9-s − 0.109·10-s + 1.00·11-s + 0.542·12-s − 0.510·13-s + 0.192·14-s − 0.258·15-s + 0.824·16-s + 0.897·17-s − 0.0815·18-s − 1.13·19-s − 0.420·20-s + 0.454·21-s − 0.246·22-s + 1.68·23-s − 0.273·24-s + 0.200·25-s + 0.124·26-s − 0.192·27-s + 0.739·28-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: no
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+3T 1 + 3T
5 1−5T 1 - 5T
29 1+29T 1 + 29T
good2 1+0.691T+8T2 1 + 0.691T + 8T^{2}
7 1+14.5T+343T2 1 + 14.5T + 343T^{2}
11 1−36.8T+1.33e3T2 1 - 36.8T + 1.33e3T^{2}
13 1+23.9T+2.19e3T2 1 + 23.9T + 2.19e3T^{2}
17 1−62.8T+4.91e3T2 1 - 62.8T + 4.91e3T^{2}
19 1+94.1T+6.85e3T2 1 + 94.1T + 6.85e3T^{2}
23 1−185.T+1.21e4T2 1 - 185.T + 1.21e4T^{2}
31 1+104.T+2.97e4T2 1 + 104.T + 2.97e4T^{2}
37 1+259.T+5.06e4T2 1 + 259.T + 5.06e4T^{2}
41 1−11.8T+6.89e4T2 1 - 11.8T + 6.89e4T^{2}
43 1+61.0T+7.95e4T2 1 + 61.0T + 7.95e4T^{2}
47 1−281.T+1.03e5T2 1 - 281.T + 1.03e5T^{2}
53 1+170.T+1.48e5T2 1 + 170.T + 1.48e5T^{2}
59 1+636.T+2.05e5T2 1 + 636.T + 2.05e5T^{2}
61 1+379.T+2.26e5T2 1 + 379.T + 2.26e5T^{2}
67 1+623.T+3.00e5T2 1 + 623.T + 3.00e5T^{2}
71 1−298.T+3.57e5T2 1 - 298.T + 3.57e5T^{2}
73 1+524.T+3.89e5T2 1 + 524.T + 3.89e5T^{2}
79 1−563.T+4.93e5T2 1 - 563.T + 4.93e5T^{2}
83 1+885.T+5.71e5T2 1 + 885.T + 5.71e5T^{2}
89 1+447.T+7.04e5T2 1 + 447.T + 7.04e5T^{2}
97 1+558.T+9.12e5T2 1 + 558.T + 9.12e5T^{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

−10.15867060504802352030157116514, −9.363192040235874905048343646027, −8.793178267373351469376739021283, −7.39441827492440061712253587055, −6.45628805481239957303103005381, −5.46920412329107535706262119468, −4.47359411889405413605184688862, −3.30012771818319658793766011066, −1.36958129817956984038366708115, 0, 1.36958129817956984038366708115, 3.30012771818319658793766011066, 4.47359411889405413605184688862, 5.46920412329107535706262119468, 6.45628805481239957303103005381, 7.39441827492440061712253587055, 8.793178267373351469376739021283, 9.363192040235874905048343646027, 10.15867060504802352030157116514

Graph of the ZZ-function along the critical line