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Properties

Label 2-10-1.1-c9-0-2
Degree 22
Conductor 1010
Sign −1-1
Analytic cond. 5.150355.15035
Root an. cond. 2.269442.26944
Motivic weight 99
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 46·3-s + 256·4-s − 625·5-s − 736·6-s − 1.03e4·7-s − 4.09e3·8-s − 1.75e4·9-s + 1.00e4·10-s − 5.56e3·11-s + 1.17e4·12-s + 4.59e4·13-s + 1.65e5·14-s − 2.87e4·15-s + 6.55e4·16-s − 3.81e5·17-s + 2.81e5·18-s + 6.10e5·19-s − 1.60e5·20-s − 4.74e5·21-s + 8.90e4·22-s − 1.44e6·23-s − 1.88e5·24-s + 3.90e5·25-s − 7.35e5·26-s − 1.71e6·27-s − 2.64e6·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.327·3-s + 1/2·4-s − 0.447·5-s − 0.231·6-s − 1.62·7-s − 0.353·8-s − 0.892·9-s + 0.316·10-s − 0.114·11-s + 0.163·12-s + 0.446·13-s + 1.14·14-s − 0.146·15-s + 1/4·16-s − 1.10·17-s + 0.631·18-s + 1.07·19-s − 0.223·20-s − 0.532·21-s + 0.0810·22-s − 1.07·23-s − 0.115·24-s + 1/5·25-s − 0.315·26-s − 0.620·27-s − 0.812·28-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=(− Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+9/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: −1-1
Analytic conductor: 5.150355.15035
Root analytic conductor: 2.269442.26944
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 10, ( :9/2), −1)(2,\ 10,\ (\ :9/2),\ -1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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+p4T 1 + p^{4} T
5 1+p4T 1 + p^{4} T
good3 1−46T+p9T2 1 - 46 T + p^{9} T^{2}
7 1+1474pT+p9T2 1 + 1474 p T + p^{9} T^{2}
11 1+5568T+p9T2 1 + 5568 T + p^{9} T^{2}
13 1−45986T+p9T2 1 - 45986 T + p^{9} T^{2}
17 1+381318T+p9T2 1 + 381318 T + p^{9} T^{2}
19 1−610460T+p9T2 1 - 610460 T + p^{9} T^{2}
23 1+1447914T+p9T2 1 + 1447914 T + p^{9} T^{2}
29 1−5385510T+p9T2 1 - 5385510 T + p^{9} T^{2}
31 1−3053852T+p9T2 1 - 3053852 T + p^{9} T^{2}
37 1−12889442T+p9T2 1 - 12889442 T + p^{9} T^{2}
41 1+33786618T+p9T2 1 + 33786618 T + p^{9} T^{2}
43 1+36886234T+p9T2 1 + 36886234 T + p^{9} T^{2}
47 1+44163798T+p9T2 1 + 44163798 T + p^{9} T^{2}
53 1−29746266T+p9T2 1 - 29746266 T + p^{9} T^{2}
59 1+65575380T+p9T2 1 + 65575380 T + p^{9} T^{2}
61 1−40183202T+p9T2 1 - 40183202 T + p^{9} T^{2}
67 1+115706158T+p9T2 1 + 115706158 T + p^{9} T^{2}
71 1+231681708T+p9T2 1 + 231681708 T + p^{9} T^{2}
73 1−358691906T+p9T2 1 - 358691906 T + p^{9} T^{2}
79 1+486017080T+p9T2 1 + 486017080 T + p^{9} T^{2}
83 1−251168886T+p9T2 1 - 251168886 T + p^{9} T^{2}
89 1+526039110T+p9T2 1 + 526039110 T + p^{9} T^{2}
97 1+1075981438T+p9T2 1 + 1075981438 T + p^{9} 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

−18.17674770234109574340073195289, −16.49760346673342694979299488176, −15.51869766232036129089545281682, −13.52884631526750982706488792743, −11.75007066646563119595358566331, −9.904625620639392787109259989547, −8.445358808011349762724740333691, −6.48036026027725390451528005228, −3.09665967626266841373478259759, 0, 3.09665967626266841373478259759, 6.48036026027725390451528005228, 8.445358808011349762724740333691, 9.904625620639392787109259989547, 11.75007066646563119595358566331, 13.52884631526750982706488792743, 15.51869766232036129089545281682, 16.49760346673342694979299488176, 18.17674770234109574340073195289

Graph of the ZZ-function along the critical line