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Properties

Label 2-88e2-1.1-c1-0-197
Degree 22
Conductor 77447744
Sign −1-1
Analytic cond. 61.836161.8361
Root an. cond. 7.863597.86359
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  + 3·3-s − 5-s + 6·9-s − 6·13-s − 3·15-s + 4·17-s − 6·19-s − 3·23-s − 4·25-s + 9·27-s − 4·29-s + 9·31-s − 7·37-s − 18·39-s + 2·41-s − 6·43-s − 6·45-s − 12·47-s − 7·49-s + 12·51-s − 2·53-s − 18·57-s + 9·59-s + 8·61-s + 6·65-s − 15·67-s − 9·69-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s + 2·9-s − 1.66·13-s − 0.774·15-s + 0.970·17-s − 1.37·19-s − 0.625·23-s − 4/5·25-s + 1.73·27-s − 0.742·29-s + 1.61·31-s − 1.15·37-s − 2.88·39-s + 0.312·41-s − 0.914·43-s − 0.894·45-s − 1.75·47-s − 49-s + 1.68·51-s − 0.274·53-s − 2.38·57-s + 1.17·59-s + 1.02·61-s + 0.744·65-s − 1.83·67-s − 1.08·69-s + ⋯

Functional equation

Λ(s)=(7744s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7744s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 77447744    =    26⋅1122^{6} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 61.836161.8361
Root analytic conductor: 7.863597.86359
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7744, ( :1/2), −1)(2,\ 7744,\ (\ :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 1
11 1 1
good3 1−pT+pT2 1 - p T + p T^{2}
5 1+T+pT2 1 + T + p T^{2}
7 1+pT2 1 + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−9T+pT2 1 - 9 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+15T+pT2 1 + 15 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+6T+pT2 1 + 6 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+5T+pT2 1 + 5 T + p T^{2}
97 1+3T+pT2 1 + 3 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

−7.77504752207978386728366476611, −7.08072343183803009576323277610, −6.37305639235248964419573519168, −5.18502090217935779773115652171, −4.46713554310869373480031428571, −3.77327816150142389344477767412, −3.09410655009033832646896370919, −2.32539615240176503621700363929, −1.67933822551948351378331307814, 0, 1.67933822551948351378331307814, 2.32539615240176503621700363929, 3.09410655009033832646896370919, 3.77327816150142389344477767412, 4.46713554310869373480031428571, 5.18502090217935779773115652171, 6.37305639235248964419573519168, 7.08072343183803009576323277610, 7.77504752207978386728366476611

Graph of the ZZ-function along the critical line