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Properties

Label 2-435-1.1-c3-0-32
Degree 22
Conductor 435435
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.88·2-s − 3·3-s + 15.8·4-s + 5·5-s − 14.6·6-s + 5.48·7-s + 38.3·8-s + 9·9-s + 24.4·10-s + 50.1·11-s − 47.5·12-s − 20.7·13-s + 26.7·14-s − 15·15-s + 60.3·16-s − 18.8·17-s + 43.9·18-s + 78.8·19-s + 79.2·20-s − 16.4·21-s + 244.·22-s − 6.33·23-s − 114.·24-s + 25·25-s − 101.·26-s − 27·27-s + 86.9·28-s + ⋯
L(s)  = 1  + 1.72·2-s − 0.577·3-s + 1.98·4-s + 0.447·5-s − 0.996·6-s + 0.296·7-s + 1.69·8-s + 0.333·9-s + 0.772·10-s + 1.37·11-s − 1.14·12-s − 0.442·13-s + 0.511·14-s − 0.258·15-s + 0.942·16-s − 0.268·17-s + 0.575·18-s + 0.951·19-s + 0.885·20-s − 0.171·21-s + 2.37·22-s − 0.0574·23-s − 0.977·24-s + 0.200·25-s − 0.763·26-s − 0.192·27-s + 0.586·28-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: 11
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: 00
Selberg data: (2, 435, ( :3/2), 1)(2,\ 435,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.3971733435.397173343
L(12)L(\frac12) ≈\approx 5.3971733435.397173343
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 1−4.88T+8T2 1 - 4.88T + 8T^{2}
7 1−5.48T+343T2 1 - 5.48T + 343T^{2}
11 1−50.1T+1.33e3T2 1 - 50.1T + 1.33e3T^{2}
13 1+20.7T+2.19e3T2 1 + 20.7T + 2.19e3T^{2}
17 1+18.8T+4.91e3T2 1 + 18.8T + 4.91e3T^{2}
19 1−78.8T+6.85e3T2 1 - 78.8T + 6.85e3T^{2}
23 1+6.33T+1.21e4T2 1 + 6.33T + 1.21e4T^{2}
31 1−310.T+2.97e4T2 1 - 310.T + 2.97e4T^{2}
37 1+338.T+5.06e4T2 1 + 338.T + 5.06e4T^{2}
41 1−353.T+6.89e4T2 1 - 353.T + 6.89e4T^{2}
43 1−507.T+7.95e4T2 1 - 507.T + 7.95e4T^{2}
47 1+112.T+1.03e5T2 1 + 112.T + 1.03e5T^{2}
53 1+144.T+1.48e5T2 1 + 144.T + 1.48e5T^{2}
59 1+342.T+2.05e5T2 1 + 342.T + 2.05e5T^{2}
61 1−357.T+2.26e5T2 1 - 357.T + 2.26e5T^{2}
67 1+183.T+3.00e5T2 1 + 183.T + 3.00e5T^{2}
71 1+594.T+3.57e5T2 1 + 594.T + 3.57e5T^{2}
73 1+622.T+3.89e5T2 1 + 622.T + 3.89e5T^{2}
79 1+1.27e3T+4.93e5T2 1 + 1.27e3T + 4.93e5T^{2}
83 1+739.T+5.71e5T2 1 + 739.T + 5.71e5T^{2}
89 1−906.T+7.04e5T2 1 - 906.T + 7.04e5T^{2}
97 1−76.0T+9.12e5T2 1 - 76.0T + 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

−11.20466149468208511437419850500, −10.11528034943950327350553961328, −9.033399839563561127542577279213, −7.43211937681196381835273539961, −6.53635187264162886694540698789, −5.85428167098803690945895068284, −4.86405504162719696594876375774, −4.10687724538625121853682746059, −2.81728917315718518384932292266, −1.39753963967844542091092476708, 1.39753963967844542091092476708, 2.81728917315718518384932292266, 4.10687724538625121853682746059, 4.86405504162719696594876375774, 5.85428167098803690945895068284, 6.53635187264162886694540698789, 7.43211937681196381835273539961, 9.033399839563561127542577279213, 10.11528034943950327350553961328, 11.20466149468208511437419850500

Graph of the ZZ-function along the critical line