Properties

Label 2-1456-1.1-c1-0-27
Degree $2$
Conductor $1456$
Sign $1$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + 3.10·3-s + 2.81·5-s + 7-s + 6.62·9-s − 3.10·11-s + 13-s + 8.72·15-s − 0.524·17-s − 0.813·19-s + 3.10·21-s − 7.33·23-s + 2.91·25-s + 11.2·27-s + 8.28·29-s − 1.39·31-s − 9.62·33-s + 2.81·35-s − 6.15·37-s + 3.10·39-s − 4.20·41-s − 6.75·43-s + 18.6·45-s + 5.97·47-s + 49-s − 1.62·51-s − 2.49·53-s − 8.72·55-s + ⋯
L(s)  = 1  + 1.79·3-s + 1.25·5-s + 0.377·7-s + 2.20·9-s − 0.935·11-s + 0.277·13-s + 2.25·15-s − 0.127·17-s − 0.186·19-s + 0.677·21-s − 1.53·23-s + 0.583·25-s + 2.16·27-s + 1.53·29-s − 0.250·31-s − 1.67·33-s + 0.475·35-s − 1.01·37-s + 0.496·39-s − 0.656·41-s − 1.03·43-s + 2.77·45-s + 0.870·47-s + 0.142·49-s − 0.227·51-s − 0.342·53-s − 1.17·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.904448448\)
\(L(\frac12)\) \(\approx\) \(3.904448448\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 - 3.10T + 3T^{2} \)
5 \( 1 - 2.81T + 5T^{2} \)
11 \( 1 + 3.10T + 11T^{2} \)
17 \( 1 + 0.524T + 17T^{2} \)
19 \( 1 + 0.813T + 19T^{2} \)
23 \( 1 + 7.33T + 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 + 1.39T + 31T^{2} \)
37 \( 1 + 6.15T + 37T^{2} \)
41 \( 1 + 4.20T + 41T^{2} \)
43 \( 1 + 6.75T + 43T^{2} \)
47 \( 1 - 5.97T + 47T^{2} \)
53 \( 1 + 2.49T + 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 8.72T + 71T^{2} \)
73 \( 1 + 2.34T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 1.18T + 97T^{2} \)
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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

−9.495481766396675463960404906458, −8.547364917649146820897717181080, −8.243895731884702410457988735007, −7.29783290257583953046814290682, −6.34009340493963960645564380591, −5.30493396736537125383925114915, −4.29190655729097811687734341500, −3.19442215475345536716127725010, −2.31185673482776118411148128145, −1.67601087289506257611219709057, 1.67601087289506257611219709057, 2.31185673482776118411148128145, 3.19442215475345536716127725010, 4.29190655729097811687734341500, 5.30493396736537125383925114915, 6.34009340493963960645564380591, 7.29783290257583953046814290682, 8.243895731884702410457988735007, 8.547364917649146820897717181080, 9.495481766396675463960404906458

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