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Properties

Label 2-666-1.1-c1-0-12
Degree 22
Conductor 666666
Sign −1-1
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
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 − 3·11-s − 13-s + 14-s + 16-s + 3·17-s − 7·19-s + 3·22-s − 3·23-s − 5·25-s + 26-s − 28-s + 2·31-s − 32-s − 3·34-s + 37-s + 7·38-s + 6·41-s − 4·43-s − 3·44-s + 3·46-s − 6·47-s − 6·49-s + 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.904·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s + 0.639·22-s − 0.625·23-s − 25-s + 0.196·26-s − 0.188·28-s + 0.359·31-s − 0.176·32-s − 0.514·34-s + 0.164·37-s + 1.13·38-s + 0.937·41-s − 0.609·43-s − 0.452·44-s + 0.442·46-s − 0.875·47-s − 6/7·49-s + 0.707·50-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: −1-1
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 666, ( :1/2), −1)(2,\ 666,\ (\ :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
37 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+7T+pT2 1 + 7 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−5T+pT2 1 - 5 T + p T^{2}
79 1−2T+pT2 1 - 2 T + p T^{2}
83 1+3T+pT2 1 + 3 T + p T^{2}
89 1−3T+pT2 1 - 3 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−10.06815972051721250592457118805, −9.339371053419225859875408435925, −8.208412645914458611430202370847, −7.73927537740876447103082520863, −6.55620004019545166157451145997, −5.78215828046120660270620191902, −4.49624163661933276807437922670, −3.14043362082132550386338477310, −1.96278840753819796229532576491, 0, 1.96278840753819796229532576491, 3.14043362082132550386338477310, 4.49624163661933276807437922670, 5.78215828046120660270620191902, 6.55620004019545166157451145997, 7.73927537740876447103082520863, 8.208412645914458611430202370847, 9.339371053419225859875408435925, 10.06815972051721250592457118805

Graph of the ZZ-function along the critical line