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Properties

Label 2-435-1.1-c3-0-22
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  + 0.529·2-s + 3·3-s − 7.71·4-s + 5·5-s + 1.58·6-s + 30.0·7-s − 8.32·8-s + 9·9-s + 2.64·10-s + 11.5·11-s − 23.1·12-s − 64.3·13-s + 15.9·14-s + 15·15-s + 57.3·16-s + 29.8·17-s + 4.76·18-s + 103.·19-s − 38.5·20-s + 90.2·21-s + 6.12·22-s − 29.0·23-s − 24.9·24-s + 25·25-s − 34.0·26-s + 27·27-s − 232.·28-s + ⋯
L(s)  = 1  + 0.187·2-s + 0.577·3-s − 0.964·4-s + 0.447·5-s + 0.108·6-s + 1.62·7-s − 0.367·8-s + 0.333·9-s + 0.0837·10-s + 0.317·11-s − 0.557·12-s − 1.37·13-s + 0.304·14-s + 0.258·15-s + 0.896·16-s + 0.426·17-s + 0.0624·18-s + 1.24·19-s − 0.431·20-s + 0.937·21-s + 0.0593·22-s − 0.263·23-s − 0.212·24-s + 0.200·25-s − 0.257·26-s + 0.192·27-s − 1.56·28-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 2.7242230602.724223060
L(12)L(\frac12) ≈\approx 2.7242230602.724223060
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 1−0.529T+8T2 1 - 0.529T + 8T^{2}
7 1−30.0T+343T2 1 - 30.0T + 343T^{2}
11 1−11.5T+1.33e3T2 1 - 11.5T + 1.33e3T^{2}
13 1+64.3T+2.19e3T2 1 + 64.3T + 2.19e3T^{2}
17 1−29.8T+4.91e3T2 1 - 29.8T + 4.91e3T^{2}
19 1−103.T+6.85e3T2 1 - 103.T + 6.85e3T^{2}
23 1+29.0T+1.21e4T2 1 + 29.0T + 1.21e4T^{2}
31 1+44.7T+2.97e4T2 1 + 44.7T + 2.97e4T^{2}
37 1−21.8T+5.06e4T2 1 - 21.8T + 5.06e4T^{2}
41 1−35.0T+6.89e4T2 1 - 35.0T + 6.89e4T^{2}
43 1−367.T+7.95e4T2 1 - 367.T + 7.95e4T^{2}
47 1−504.T+1.03e5T2 1 - 504.T + 1.03e5T^{2}
53 1−290.T+1.48e5T2 1 - 290.T + 1.48e5T^{2}
59 1−168.T+2.05e5T2 1 - 168.T + 2.05e5T^{2}
61 1+287.T+2.26e5T2 1 + 287.T + 2.26e5T^{2}
67 1−388.T+3.00e5T2 1 - 388.T + 3.00e5T^{2}
71 1−337.T+3.57e5T2 1 - 337.T + 3.57e5T^{2}
73 1+487.T+3.89e5T2 1 + 487.T + 3.89e5T^{2}
79 1−625.T+4.93e5T2 1 - 625.T + 4.93e5T^{2}
83 1−747.T+5.71e5T2 1 - 747.T + 5.71e5T^{2}
89 1+205.T+7.04e5T2 1 + 205.T + 7.04e5T^{2}
97 1+1.08e3T+9.12e5T2 1 + 1.08e3T + 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

−10.57332805094497608685072651851, −9.618239565442778688380682482095, −9.009451725558400980641770977491, −7.963764304028980761829979869051, −7.37293294718435019986943380447, −5.57565102745898495928118912989, −4.92462915062139852645189310350, −3.95257316214826927749233417066, −2.45498459206126874728474010141, −1.10141587077213584111898626252, 1.10141587077213584111898626252, 2.45498459206126874728474010141, 3.95257316214826927749233417066, 4.92462915062139852645189310350, 5.57565102745898495928118912989, 7.37293294718435019986943380447, 7.963764304028980761829979869051, 9.009451725558400980641770977491, 9.618239565442778688380682482095, 10.57332805094497608685072651851

Graph of the ZZ-function along the critical line