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Properties

Label 2-99-1.1-c1-0-2
Degree 22
Conductor 9999
Sign 11
Analytic cond. 0.7905180.790518
Root an. cond. 0.8891110.889111
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 5-s − 2·7-s − 2·10-s − 11-s + 4·13-s − 4·14-s − 4·16-s + 2·17-s − 2·20-s − 2·22-s + 23-s − 4·25-s + 8·26-s − 4·28-s + 7·31-s − 8·32-s + 4·34-s + 2·35-s + 3·37-s + 8·41-s − 6·43-s − 2·44-s + 2·46-s − 8·47-s − 3·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s − 0.632·10-s − 0.301·11-s + 1.10·13-s − 1.06·14-s − 16-s + 0.485·17-s − 0.447·20-s − 0.426·22-s + 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.755·28-s + 1.25·31-s − 1.41·32-s + 0.685·34-s + 0.338·35-s + 0.493·37-s + 1.24·41-s − 0.914·43-s − 0.301·44-s + 0.294·46-s − 1.16·47-s − 3/7·49-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: 11
Analytic conductor: 0.7905180.790518
Root analytic conductor: 0.8891110.889111
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 99, ( :1/2), 1)(2,\ 99,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6844963321.684496332
L(12)L(\frac12) ≈\approx 1.6844963321.684496332
L(32)L(\frac{3}{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 1
11 1+T 1 + T
good2 1−pT+pT2 1 - p T + p T^{2}
5 1+T+pT2 1 + T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+5T+pT2 1 + 5 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1+7T+pT2 1 + 7 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1+7T+pT2 1 + 7 T + p 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

−13.68040274047176229306789941909, −13.05532878430427737685314941779, −12.07025537765623530596257929043, −11.14136051051457299622132768480, −9.701226498296212058900989539920, −8.228854358433494441874279371750, −6.66503987304844816686807720890, −5.66407660024727617127120629681, −4.19085115174357929506261869806, −3.09000916592887094247460325417, 3.09000916592887094247460325417, 4.19085115174357929506261869806, 5.66407660024727617127120629681, 6.66503987304844816686807720890, 8.228854358433494441874279371750, 9.701226498296212058900989539920, 11.14136051051457299622132768480, 12.07025537765623530596257929043, 13.05532878430427737685314941779, 13.68040274047176229306789941909

Graph of the ZZ-function along the critical line