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Properties

Label 2-435-1.1-c3-0-18
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  − 3.57·2-s − 3·3-s + 4.79·4-s + 5·5-s + 10.7·6-s + 15.0·7-s + 11.4·8-s + 9·9-s − 17.8·10-s + 70.8·11-s − 14.3·12-s + 62.2·13-s − 53.7·14-s − 15·15-s − 79.3·16-s − 67.1·17-s − 32.1·18-s + 118.·19-s + 23.9·20-s − 45.0·21-s − 253.·22-s + 72.5·23-s − 34.3·24-s + 25·25-s − 222.·26-s − 27·27-s + 72.0·28-s + ⋯
L(s)  = 1  − 1.26·2-s − 0.577·3-s + 0.599·4-s + 0.447·5-s + 0.730·6-s + 0.811·7-s + 0.506·8-s + 0.333·9-s − 0.565·10-s + 1.94·11-s − 0.346·12-s + 1.32·13-s − 1.02·14-s − 0.258·15-s − 1.24·16-s − 0.958·17-s − 0.421·18-s + 1.43·19-s + 0.268·20-s − 0.468·21-s − 2.45·22-s + 0.657·23-s − 0.292·24-s + 0.200·25-s − 1.67·26-s − 0.192·27-s + 0.486·28-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 1.1929495261.192949526
L(12)L(\frac12) ≈\approx 1.1929495261.192949526
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+3.57T+8T2 1 + 3.57T + 8T^{2}
7 1−15.0T+343T2 1 - 15.0T + 343T^{2}
11 1−70.8T+1.33e3T2 1 - 70.8T + 1.33e3T^{2}
13 1−62.2T+2.19e3T2 1 - 62.2T + 2.19e3T^{2}
17 1+67.1T+4.91e3T2 1 + 67.1T + 4.91e3T^{2}
19 1−118.T+6.85e3T2 1 - 118.T + 6.85e3T^{2}
23 1−72.5T+1.21e4T2 1 - 72.5T + 1.21e4T^{2}
31 1+180.T+2.97e4T2 1 + 180.T + 2.97e4T^{2}
37 1+47.8T+5.06e4T2 1 + 47.8T + 5.06e4T^{2}
41 1−371.T+6.89e4T2 1 - 371.T + 6.89e4T^{2}
43 1+409.T+7.95e4T2 1 + 409.T + 7.95e4T^{2}
47 1−125.T+1.03e5T2 1 - 125.T + 1.03e5T^{2}
53 1+215.T+1.48e5T2 1 + 215.T + 1.48e5T^{2}
59 1−356.T+2.05e5T2 1 - 356.T + 2.05e5T^{2}
61 1+466.T+2.26e5T2 1 + 466.T + 2.26e5T^{2}
67 1+578.T+3.00e5T2 1 + 578.T + 3.00e5T^{2}
71 1−870.T+3.57e5T2 1 - 870.T + 3.57e5T^{2}
73 1−411.T+3.89e5T2 1 - 411.T + 3.89e5T^{2}
79 1−1.12e3T+4.93e5T2 1 - 1.12e3T + 4.93e5T^{2}
83 1−1.07e3T+5.71e5T2 1 - 1.07e3T + 5.71e5T^{2}
89 1−388.T+7.04e5T2 1 - 388.T + 7.04e5T^{2}
97 1+1.52e3T+9.12e5T2 1 + 1.52e3T + 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

−10.92505369369704229975299595250, −9.492825604155745370908771682853, −9.142342241576720618126581583283, −8.207811901967754044887409949826, −7.05846458998817516806568281796, −6.30348231940750811554434594808, −5.01200174110809291543161844425, −3.84962968502172973922307146854, −1.64845617539078381701702313123, −1.00545073043389954214883431763, 1.00545073043389954214883431763, 1.64845617539078381701702313123, 3.84962968502172973922307146854, 5.01200174110809291543161844425, 6.30348231940750811554434594808, 7.05846458998817516806568281796, 8.207811901967754044887409949826, 9.142342241576720618126581583283, 9.492825604155745370908771682853, 10.92505369369704229975299595250

Graph of the ZZ-function along the critical line