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Properties

Label 2-1007-1.1-c1-0-75
Degree 22
Conductor 10071007
Sign −1-1
Analytic cond. 8.040938.04093
Root an. cond. 2.835652.83565
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 3·5-s − 7-s − 3·9-s − 6·10-s + 3·11-s − 2·14-s − 4·16-s − 3·17-s − 6·18-s − 19-s − 6·20-s + 6·22-s − 8·23-s + 4·25-s − 2·28-s + 2·29-s − 2·31-s − 8·32-s − 6·34-s + 3·35-s − 6·36-s − 4·37-s − 2·38-s + 6·41-s − 5·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.34·5-s − 0.377·7-s − 9-s − 1.89·10-s + 0.904·11-s − 0.534·14-s − 16-s − 0.727·17-s − 1.41·18-s − 0.229·19-s − 1.34·20-s + 1.27·22-s − 1.66·23-s + 4/5·25-s − 0.377·28-s + 0.371·29-s − 0.359·31-s − 1.41·32-s − 1.02·34-s + 0.507·35-s − 36-s − 0.657·37-s − 0.324·38-s + 0.937·41-s − 0.762·43-s + ⋯

Functional equation

Λ(s)=(1007s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1007s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10071007    =    19⋅5319 \cdot 53
Sign: −1-1
Analytic conductor: 8.040938.04093
Root analytic conductor: 2.835652.83565
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1007, ( :1/2), −1)(2,\ 1007,\ (\ :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(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad19 1+T 1 + T
53 1+T 1 + T
good2 1−pT+pT2 1 - p T + p T^{2}
3 1+pT2 1 + p T^{2}
5 1+3T+pT2 1 + 3 T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1−13T+pT2 1 - 13 T + p T^{2}
59 1+14T+pT2 1 + 14 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−16T+pT2 1 - 16 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−9.433548470637644072669686071348, −8.626470330066807396264615662716, −7.78997306487226182646384131671, −6.65430570134456255086455486851, −6.09174406445201527513865362791, −5.00007502799761395288419220728, −3.98790135972688185122818775592, −3.62666575170944973731750940171, −2.43502883288959516917088648342, 0, 2.43502883288959516917088648342, 3.62666575170944973731750940171, 3.98790135972688185122818775592, 5.00007502799761395288419220728, 6.09174406445201527513865362791, 6.65430570134456255086455486851, 7.78997306487226182646384131671, 8.626470330066807396264615662716, 9.433548470637644072669686071348

Graph of the ZZ-function along the critical line