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Properties

Label 2-882-1.1-c5-0-40
Degree 22
Conductor 882882
Sign 11
Analytic cond. 141.458141.458
Root an. cond. 11.893611.8936
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·4-s + 86·5-s + 64·8-s + 344·10-s − 34·11-s + 3·13-s + 256·16-s − 1.90e3·17-s + 1.48e3·19-s + 1.37e3·20-s − 136·22-s + 224·23-s + 4.27e3·25-s + 12·26-s + 6.50e3·29-s − 1.73e3·31-s + 1.02e3·32-s − 7.61e3·34-s − 7.63e3·37-s + 5.95e3·38-s + 5.50e3·40-s + 1.54e4·41-s + 1.84e4·43-s − 544·44-s + 896·46-s + 1.84e4·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.53·5-s + 0.353·8-s + 1.08·10-s − 0.0847·11-s + 0.00492·13-s + 1/4·16-s − 1.59·17-s + 0.946·19-s + 0.769·20-s − 0.0599·22-s + 0.0882·23-s + 1.36·25-s + 0.00348·26-s + 1.43·29-s − 0.323·31-s + 0.176·32-s − 1.12·34-s − 0.916·37-s + 0.669·38-s + 0.543·40-s + 1.43·41-s + 1.52·43-s − 0.0423·44-s + 0.0624·46-s + 1.21·47-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 141.458141.458
Root analytic conductor: 11.893611.8936
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 882, ( :5/2), 1)(2,\ 882,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 5.3796723445.379672344
L(12)L(\frac12) ≈\approx 5.3796723445.379672344
L(72)L(\frac{7}{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)
bad2 1−p2T 1 - p^{2} T
3 1 1
7 1 1
good5 1−86T+p5T2 1 - 86 T + p^{5} T^{2}
11 1+34T+p5T2 1 + 34 T + p^{5} T^{2}
13 1−3T+p5T2 1 - 3 T + p^{5} T^{2}
17 1+112pT+p5T2 1 + 112 p T + p^{5} T^{2}
19 1−1489T+p5T2 1 - 1489 T + p^{5} T^{2}
23 1−224T+p5T2 1 - 224 T + p^{5} T^{2}
29 1−6508T+p5T2 1 - 6508 T + p^{5} T^{2}
31 1+1731T+p5T2 1 + 1731 T + p^{5} T^{2}
37 1+7633T+p5T2 1 + 7633 T + p^{5} T^{2}
41 1−15414T+p5T2 1 - 15414 T + p^{5} T^{2}
43 1−18491T+p5T2 1 - 18491 T + p^{5} T^{2}
47 1−18462T+p5T2 1 - 18462 T + p^{5} T^{2}
53 1−19956T+p5T2 1 - 19956 T + p^{5} T^{2}
59 1+31828T+p5T2 1 + 31828 T + p^{5} T^{2}
61 1−57654T+p5T2 1 - 57654 T + p^{5} T^{2}
67 1+60563T+p5T2 1 + 60563 T + p^{5} T^{2}
71 1−44834T+p5T2 1 - 44834 T + p^{5} T^{2}
73 1+20821T+p5T2 1 + 20821 T + p^{5} T^{2}
79 1+30531T+p5T2 1 + 30531 T + p^{5} T^{2}
83 1−110602T+p5T2 1 - 110602 T + p^{5} T^{2}
89 1+58992T+p5T2 1 + 58992 T + p^{5} T^{2}
97 1−119846T+p5T2 1 - 119846 T + p^{5} T^{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

−9.381505732498017334352561443603, −8.781340715330865038513138619024, −7.42732619370093528572541300681, −6.57511890994112369420462263220, −5.86307452314953997834974922781, −5.10660074364078447294055615182, −4.15683984339772916483016734544, −2.76252317727516102124637699455, −2.13088633506956037150618080989, −0.958677653623058506280910806501, 0.958677653623058506280910806501, 2.13088633506956037150618080989, 2.76252317727516102124637699455, 4.15683984339772916483016734544, 5.10660074364078447294055615182, 5.86307452314953997834974922781, 6.57511890994112369420462263220, 7.42732619370093528572541300681, 8.781340715330865038513138619024, 9.381505732498017334352561443603

Graph of the ZZ-function along the critical line