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Properties

Label 2-4050-1.1-c1-0-48
Degree 22
Conductor 40504050
Sign −1-1
Analytic cond. 32.339432.3394
Root an. cond. 5.686775.68677
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 + 4-s − 7-s − 8-s + 2·11-s − 6·13-s + 14-s + 16-s + 2·17-s + 6·19-s − 2·22-s − 23-s + 6·26-s − 28-s − 9·29-s − 2·31-s − 32-s − 2·34-s + 2·37-s − 6·38-s + 11·41-s − 4·43-s + 2·44-s + 46-s + 7·47-s − 6·49-s − 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.603·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 1.37·19-s − 0.426·22-s − 0.208·23-s + 1.17·26-s − 0.188·28-s − 1.67·29-s − 0.359·31-s − 0.176·32-s − 0.342·34-s + 0.328·37-s − 0.973·38-s + 1.71·41-s − 0.609·43-s + 0.301·44-s + 0.147·46-s + 1.02·47-s − 6/7·49-s − 0.832·52-s + ⋯

Functional equation

Λ(s)=(4050s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4050s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40504050    =    2⋅34⋅522 \cdot 3^{4} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 32.339432.3394
Root analytic conductor: 5.686775.68677
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4050, ( :1/2), −1)(2,\ 4050,\ (\ :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 1
5 1 1
good7 1+T+pT2 1 + T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+T+pT2 1 + T + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−11T+pT2 1 - 11 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−7T+pT2 1 - 7 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1+11T+pT2 1 + 11 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1−11T+pT2 1 - 11 T + p T^{2}
89 1+T+pT2 1 + T + p T^{2}
97 1+8T+pT2 1 + 8 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

−7.83363987662291462133290912027, −7.51054612699914631844435303243, −6.85375087117219460892529582818, −5.86969935881180331945520030129, −5.25833616399397912417664210182, −4.18242956185978364798445194529, −3.23108286709011396574644230445, −2.39873837184369674455978706927, −1.29732355983093497181986455091, 0, 1.29732355983093497181986455091, 2.39873837184369674455978706927, 3.23108286709011396574644230445, 4.18242956185978364798445194529, 5.25833616399397912417664210182, 5.86969935881180331945520030129, 6.85375087117219460892529582818, 7.51054612699914631844435303243, 7.83363987662291462133290912027

Graph of the ZZ-function along the critical line