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Properties

Label 2-26-1.1-c3-0-1
Degree 22
Conductor 2626
Sign 11
Analytic cond. 1.534041.53404
Root an. cond. 1.238561.23856
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 4·4-s + 17·5-s − 2·6-s − 35·7-s + 8·8-s − 26·9-s + 34·10-s + 2·11-s − 4·12-s + 13·13-s − 70·14-s − 17·15-s + 16·16-s − 19·17-s − 52·18-s + 94·19-s + 68·20-s + 35·21-s + 4·22-s − 72·23-s − 8·24-s + 164·25-s + 26·26-s + 53·27-s − 140·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.192·3-s + 1/2·4-s + 1.52·5-s − 0.136·6-s − 1.88·7-s + 0.353·8-s − 0.962·9-s + 1.07·10-s + 0.0548·11-s − 0.0962·12-s + 0.277·13-s − 1.33·14-s − 0.292·15-s + 1/4·16-s − 0.271·17-s − 0.680·18-s + 1.13·19-s + 0.760·20-s + 0.363·21-s + 0.0387·22-s − 0.652·23-s − 0.0680·24-s + 1.31·25-s + 0.196·26-s + 0.377·27-s − 0.944·28-s + ⋯

Functional equation

Λ(s)=(26s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(26s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2626    =    2⋅132 \cdot 13
Sign: 11
Analytic conductor: 1.534041.53404
Root analytic conductor: 1.238561.23856
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 26, ( :3/2), 1)(2,\ 26,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.6098123541.609812354
L(12)L(\frac12) ≈\approx 1.6098123541.609812354
L(52)L(\frac{5}{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−pT 1 - p T
13 1−pT 1 - p T
good3 1+T+p3T2 1 + T + p^{3} T^{2}
5 1−17T+p3T2 1 - 17 T + p^{3} T^{2}
7 1+5pT+p3T2 1 + 5 p T + p^{3} T^{2}
11 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
17 1+19T+p3T2 1 + 19 T + p^{3} T^{2}
19 1−94T+p3T2 1 - 94 T + p^{3} T^{2}
23 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
29 1−246T+p3T2 1 - 246 T + p^{3} T^{2}
31 1+100T+p3T2 1 + 100 T + p^{3} T^{2}
37 1+11T+p3T2 1 + 11 T + p^{3} T^{2}
41 1+280T+p3T2 1 + 280 T + p^{3} T^{2}
43 1−241T+p3T2 1 - 241 T + p^{3} T^{2}
47 1−137T+p3T2 1 - 137 T + p^{3} T^{2}
53 1+232T+p3T2 1 + 232 T + p^{3} T^{2}
59 1+386T+p3T2 1 + 386 T + p^{3} T^{2}
61 1−64T+p3T2 1 - 64 T + p^{3} T^{2}
67 1+10pT+p3T2 1 + 10 p T + p^{3} T^{2}
71 1−55T+p3T2 1 - 55 T + p^{3} T^{2}
73 1+838T+p3T2 1 + 838 T + p^{3} T^{2}
79 1−1016T+p3T2 1 - 1016 T + p^{3} T^{2}
83 1−420T+p3T2 1 - 420 T + p^{3} T^{2}
89 1+934T+p3T2 1 + 934 T + p^{3} T^{2}
97 1+1154T+p3T2 1 + 1154 T + p^{3} 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

−16.80437786070971894766043817922, −15.85286329662653664585933001688, −14.05562219357208349848035310802, −13.40994395178736909586100221211, −12.19007633631656607136146195801, −10.35868729698139241101929943743, −9.261113333992029619217116508410, −6.52009248057915953419005493342, −5.68975293420522781236146597816, −2.95389794381125198681312522075, 2.95389794381125198681312522075, 5.68975293420522781236146597816, 6.52009248057915953419005493342, 9.261113333992029619217116508410, 10.35868729698139241101929943743, 12.19007633631656607136146195801, 13.40994395178736909586100221211, 14.05562219357208349848035310802, 15.85286329662653664585933001688, 16.80437786070971894766043817922

Graph of the ZZ-function along the critical line