Properties

Label 2-1336-1.1-c1-0-3
Degree $2$
Conductor $1336$
Sign $1$
Analytic cond. $10.6680$
Root an. cond. $3.26619$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.26·3-s − 0.339·5-s − 4.08·7-s − 1.39·9-s + 2.81·11-s − 3.28·13-s + 0.430·15-s − 3.05·17-s + 6.09·19-s + 5.17·21-s + 0.204·23-s − 4.88·25-s + 5.56·27-s − 2.58·29-s + 7.00·31-s − 3.56·33-s + 1.38·35-s + 2.47·37-s + 4.16·39-s + 4.44·41-s + 8.73·43-s + 0.472·45-s − 2.38·47-s + 9.67·49-s + 3.87·51-s + 1.43·53-s − 0.954·55-s + ⋯
L(s)  = 1  − 0.732·3-s − 0.151·5-s − 1.54·7-s − 0.463·9-s + 0.847·11-s − 0.910·13-s + 0.111·15-s − 0.740·17-s + 1.39·19-s + 1.12·21-s + 0.0425·23-s − 0.976·25-s + 1.07·27-s − 0.479·29-s + 1.25·31-s − 0.620·33-s + 0.234·35-s + 0.407·37-s + 0.666·39-s + 0.694·41-s + 1.33·43-s + 0.0704·45-s − 0.348·47-s + 1.38·49-s + 0.542·51-s + 0.196·53-s − 0.128·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1336\)    =    \(2^{3} \cdot 167\)
Sign: $1$
Analytic conductor: \(10.6680\)
Root analytic conductor: \(3.26619\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1336,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7521820019\)
\(L(\frac12)\) \(\approx\) \(0.7521820019\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 - T \)
good3 \( 1 + 1.26T + 3T^{2} \)
5 \( 1 + 0.339T + 5T^{2} \)
7 \( 1 + 4.08T + 7T^{2} \)
11 \( 1 - 2.81T + 11T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 - 6.09T + 19T^{2} \)
23 \( 1 - 0.204T + 23T^{2} \)
29 \( 1 + 2.58T + 29T^{2} \)
31 \( 1 - 7.00T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 - 4.44T + 41T^{2} \)
43 \( 1 - 8.73T + 43T^{2} \)
47 \( 1 + 2.38T + 47T^{2} \)
53 \( 1 - 1.43T + 53T^{2} \)
59 \( 1 - 5.48T + 59T^{2} \)
61 \( 1 + 5.24T + 61T^{2} \)
67 \( 1 + 9.65T + 67T^{2} \)
71 \( 1 + 1.65T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 6.14T + 79T^{2} \)
83 \( 1 - 1.56T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 2.47T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.499065121970245825467000444204, −9.182235642739005870652929026992, −7.86591397493081278363450346158, −6.97238672596746600875212803764, −6.27955920371159052478084984410, −5.64562593628741481572171567579, −4.53967827121825141249346459736, −3.48857561930286863394113977217, −2.54869721513187973697415170072, −0.62905087182049633332248604445, 0.62905087182049633332248604445, 2.54869721513187973697415170072, 3.48857561930286863394113977217, 4.53967827121825141249346459736, 5.64562593628741481572171567579, 6.27955920371159052478084984410, 6.97238672596746600875212803764, 7.86591397493081278363450346158, 9.182235642739005870652929026992, 9.499065121970245825467000444204

Graph of the $Z$-function along the critical line