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Properties

Label 2-8850-1.1-c1-0-130
Degree 22
Conductor 88508850
Sign −1-1
Analytic cond. 70.667670.6676
Root an. cond. 8.406408.40640
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-s − 3-s + 4-s + 6-s + 3·7-s − 8-s + 9-s − 3·11-s − 12-s + 5·13-s − 3·14-s + 16-s − 17-s − 18-s − 2·19-s − 3·21-s + 3·22-s + 24-s − 5·26-s − 27-s + 3·28-s − 6·29-s − 2·31-s − 32-s + 3·33-s + 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.38·13-s − 0.801·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.654·21-s + 0.639·22-s + 0.204·24-s − 0.980·26-s − 0.192·27-s + 0.566·28-s − 1.11·29-s − 0.359·31-s − 0.176·32-s + 0.522·33-s + 0.171·34-s + 1/6·36-s + ⋯

Functional equation

Λ(s)=(8850s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8850s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 88508850    =    2⋅3⋅52⋅592 \cdot 3 \cdot 5^{2} \cdot 59
Sign: −1-1
Analytic conductor: 70.667670.6676
Root analytic conductor: 8.406408.40640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8850, ( :1/2), −1)(2,\ 8850,\ (\ :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)
bad2 1+T 1 + T
3 1+T 1 + T
5 1 1
59 1+T 1 + T
good7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1+T+pT2 1 + T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1−T+pT2 1 - T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+9T+pT2 1 + 9 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−T+pT2 1 - T + p T^{2}
83 1+15T+pT2 1 + 15 T + p T^{2}
89 1+16T+pT2 1 + 16 T + p T^{2}
97 1+pT2 1 + 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

−7.51641777061424011766017254141, −6.85968241202142381330201694430, −5.88267173286614013533134098576, −5.61803826822241723304384507825, −4.63080403494839023271633987682, −3.98287875236199371416176310655, −2.88320061980396347599630187844, −1.88523665772311216717857407774, −1.22175888953010540094546470927, 0, 1.22175888953010540094546470927, 1.88523665772311216717857407774, 2.88320061980396347599630187844, 3.98287875236199371416176310655, 4.63080403494839023271633987682, 5.61803826822241723304384507825, 5.88267173286614013533134098576, 6.85968241202142381330201694430, 7.51641777061424011766017254141

Graph of the ZZ-function along the critical line