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Properties

Label 2-1098-1.1-c1-0-7
Degree 22
Conductor 10981098
Sign 11
Analytic cond. 8.767578.76757
Root an. cond. 2.961002.96100
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-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 6·11-s − 6·13-s + 4·14-s + 16-s + 3·17-s − 6·19-s − 20-s + 6·22-s + 7·23-s − 4·25-s − 6·26-s + 4·28-s + 6·31-s + 32-s + 3·34-s − 4·35-s − 3·37-s − 6·38-s − 40-s + 7·43-s + 6·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.80·11-s − 1.66·13-s + 1.06·14-s + 1/4·16-s + 0.727·17-s − 1.37·19-s − 0.223·20-s + 1.27·22-s + 1.45·23-s − 4/5·25-s − 1.17·26-s + 0.755·28-s + 1.07·31-s + 0.176·32-s + 0.514·34-s − 0.676·35-s − 0.493·37-s − 0.973·38-s − 0.158·40-s + 1.06·43-s + 0.904·44-s + ⋯

Functional equation

Λ(s)=(1098s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1098s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10981098    =    2⋅32⋅612 \cdot 3^{2} \cdot 61
Sign: 11
Analytic conductor: 8.767578.76757
Root analytic conductor: 2.961002.96100
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1098, ( :1/2), 1)(2,\ 1098,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8563517412.856351741
L(12)L(\frac12) ≈\approx 2.8563517412.856351741
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)
bad2 1−T 1 - T
3 1 1
61 1+T 1 + T
good5 1+T+pT2 1 + T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1−7T+pT2 1 - 7 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−7T+pT2 1 - 7 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1−3T+pT2 1 - 3 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1+15T+pT2 1 + 15 T + p T^{2}
89 1+9T+pT2 1 + 9 T + p T^{2}
97 1+13T+pT2 1 + 13 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

−9.917461859219297653369521903530, −8.954461847508739444883118593331, −8.069357192077553612030161472406, −7.30055553795352770723922990587, −6.53709602046399974447217609868, −5.34558974721025801567566921767, −4.54876985837096746161070305611, −3.95885180382732927496948309529, −2.52097089588598704365976417676, −1.35091118273881221529817713534, 1.35091118273881221529817713534, 2.52097089588598704365976417676, 3.95885180382732927496948309529, 4.54876985837096746161070305611, 5.34558974721025801567566921767, 6.53709602046399974447217609868, 7.30055553795352770723922990587, 8.069357192077553612030161472406, 8.954461847508739444883118593331, 9.917461859219297653369521903530

Graph of the ZZ-function along the critical line