Properties

Label 2-82110-1.1-c1-0-48
Degree 22
Conductor 8211082110
Sign 11
Analytic cond. 655.651655.651
Root an. cond. 25.605625.6056
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 2·11-s + 12-s − 4·13-s + 14-s − 15-s + 16-s + 17-s − 18-s − 4·19-s − 20-s − 21-s + 2·22-s + 23-s − 24-s + 25-s + 4·26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(82110s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(82110s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 82110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8211082110    =    235717232 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 23
Sign: 11
Analytic conductor: 655.651655.651
Root analytic conductor: 25.605625.6056
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 82110, ( :1/2), 1)(2,\ 82110,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+T 1 + T
3 1T 1 - T
5 1+T 1 + T
7 1+T 1 + T
17 1T 1 - T
23 1T 1 - T
good11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
29 12T+pT2 1 - 2 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 16T+pT2 1 - 6 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1+14T+pT2 1 + 14 T + p T^{2}
83 112T+pT2 1 - 12 T + p T^{2}
89 114T+pT2 1 - 14 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p T^{2}
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   L(s)=p j=12(1αj,pps)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

−14.73662688019337, −14.03112344402860, −13.42820260668080, −12.86921767799434, −12.50216768525979, −11.99693718085330, −11.49069498749048, −10.77246670865125, −10.37374190353781, −9.998388645427815, −9.359095179278379, −8.966058182120747, −8.399183971531302, −7.781807747701441, −7.647576348448239, −6.823785039332778, −6.549064845176256, −5.776509903600139, −4.916967684710496, −4.672994562754286, −3.742656126088552, −3.126693812592978, −2.745012539146785, −1.949608857622513, −1.382177897246375, 0, 0, 1.382177897246375, 1.949608857622513, 2.745012539146785, 3.126693812592978, 3.742656126088552, 4.672994562754286, 4.916967684710496, 5.776509903600139, 6.549064845176256, 6.823785039332778, 7.647576348448239, 7.781807747701441, 8.399183971531302, 8.966058182120747, 9.359095179278379, 9.998388645427815, 10.37374190353781, 10.77246670865125, 11.49069498749048, 11.99693718085330, 12.50216768525979, 12.86921767799434, 13.42820260668080, 14.03112344402860, 14.73662688019337

Graph of the ZZ-function along the critical line