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Properties

Label 2-435-1.1-c3-0-50
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  + 4.42·2-s − 3·3-s + 11.6·4-s − 5·5-s − 13.2·6-s − 2.85·7-s + 15.9·8-s + 9·9-s − 22.1·10-s + 6.18·11-s − 34.8·12-s − 56.2·13-s − 12.6·14-s + 15·15-s − 22.2·16-s − 46.4·17-s + 39.8·18-s − 156.·19-s − 58.0·20-s + 8.56·21-s + 27.3·22-s + 154.·23-s − 47.8·24-s + 25·25-s − 249.·26-s − 27·27-s − 33.1·28-s + ⋯
L(s)  = 1  + 1.56·2-s − 0.577·3-s + 1.45·4-s − 0.447·5-s − 0.903·6-s − 0.154·7-s + 0.704·8-s + 0.333·9-s − 0.700·10-s + 0.169·11-s − 0.837·12-s − 1.20·13-s − 0.241·14-s + 0.258·15-s − 0.347·16-s − 0.663·17-s + 0.521·18-s − 1.88·19-s − 0.648·20-s + 0.0889·21-s + 0.265·22-s + 1.40·23-s − 0.406·24-s + 0.200·25-s − 1.87·26-s − 0.192·27-s − 0.223·28-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 1−4.42T+8T2 1 - 4.42T + 8T^{2}
7 1+2.85T+343T2 1 + 2.85T + 343T^{2}
11 1−6.18T+1.33e3T2 1 - 6.18T + 1.33e3T^{2}
13 1+56.2T+2.19e3T2 1 + 56.2T + 2.19e3T^{2}
17 1+46.4T+4.91e3T2 1 + 46.4T + 4.91e3T^{2}
19 1+156.T+6.85e3T2 1 + 156.T + 6.85e3T^{2}
23 1−154.T+1.21e4T2 1 - 154.T + 1.21e4T^{2}
31 1−29.4T+2.97e4T2 1 - 29.4T + 2.97e4T^{2}
37 1+45.2T+5.06e4T2 1 + 45.2T + 5.06e4T^{2}
41 1+61.9T+6.89e4T2 1 + 61.9T + 6.89e4T^{2}
43 1+378.T+7.95e4T2 1 + 378.T + 7.95e4T^{2}
47 1+488.T+1.03e5T2 1 + 488.T + 1.03e5T^{2}
53 1−77.0T+1.48e5T2 1 - 77.0T + 1.48e5T^{2}
59 1+770.T+2.05e5T2 1 + 770.T + 2.05e5T^{2}
61 1−598.T+2.26e5T2 1 - 598.T + 2.26e5T^{2}
67 1−500.T+3.00e5T2 1 - 500.T + 3.00e5T^{2}
71 1−1.04e3T+3.57e5T2 1 - 1.04e3T + 3.57e5T^{2}
73 1−790.T+3.89e5T2 1 - 790.T + 3.89e5T^{2}
79 1−1.03e3T+4.93e5T2 1 - 1.03e3T + 4.93e5T^{2}
83 1+308.T+5.71e5T2 1 + 308.T + 5.71e5T^{2}
89 1−336.T+7.04e5T2 1 - 336.T + 7.04e5T^{2}
97 1+590.T+9.12e5T2 1 + 590.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

−10.77487000215626732074046570962, −9.521842931072664569131563623075, −8.271399650551271228569403527726, −6.81847376259381468660484553122, −6.51789751958666533661370241258, −5.07694692134468644649625515065, −4.62160497518057498119575018904, −3.48150455760780017226686027191, −2.21877362611530292879105166749, 0, 2.21877362611530292879105166749, 3.48150455760780017226686027191, 4.62160497518057498119575018904, 5.07694692134468644649625515065, 6.51789751958666533661370241258, 6.81847376259381468660484553122, 8.271399650551271228569403527726, 9.521842931072664569131563623075, 10.77487000215626732074046570962

Graph of the ZZ-function along the critical line