Properties

Label 2-704-1.1-c1-0-18
Degree $2$
Conductor $704$
Sign $-1$
Analytic cond. $5.62146$
Root an. cond. $2.37096$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·3-s − 3.56·5-s − 0.561·9-s − 11-s − 2·13-s − 5.56·15-s − 1.12·17-s − 7.12·19-s + 4.68·23-s + 7.68·25-s − 5.56·27-s + 1.12·29-s − 9.56·31-s − 1.56·33-s − 6.68·37-s − 3.12·39-s + 8.24·41-s − 7.12·43-s + 2·45-s − 4·47-s − 7·49-s − 1.75·51-s + 8.24·53-s + 3.56·55-s − 11.1·57-s + 12.6·59-s + 15.3·61-s + ⋯
L(s)  = 1  + 0.901·3-s − 1.59·5-s − 0.187·9-s − 0.301·11-s − 0.554·13-s − 1.43·15-s − 0.272·17-s − 1.63·19-s + 0.976·23-s + 1.53·25-s − 1.07·27-s + 0.208·29-s − 1.71·31-s − 0.271·33-s − 1.09·37-s − 0.500·39-s + 1.28·41-s − 1.08·43-s + 0.298·45-s − 0.583·47-s − 49-s − 0.245·51-s + 1.13·53-s + 0.480·55-s − 1.47·57-s + 1.65·59-s + 1.96·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-1$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 1.56T + 3T^{2} \)
5 \( 1 + 3.56T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 - 4.68T + 23T^{2} \)
29 \( 1 - 1.12T + 29T^{2} \)
31 \( 1 + 9.56T + 31T^{2} \)
37 \( 1 + 6.68T + 37T^{2} \)
41 \( 1 - 8.24T + 41T^{2} \)
43 \( 1 + 7.12T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 - 4.68T + 67T^{2} \)
71 \( 1 - 3.31T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 4.87T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 3.56T + 89T^{2} \)
97 \( 1 + 6.68T + 97T^{2} \)
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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

−9.965088507638647073968732990367, −8.656929244624234630824060750928, −8.530945589208049100672885995963, −7.49674853250841138701759634638, −6.84666074039592107350230004766, −5.28888958868363812420471932101, −4.17676354055744813843210672677, −3.41502641256963178080103047658, −2.30906917918481945396602919838, 0, 2.30906917918481945396602919838, 3.41502641256963178080103047658, 4.17676354055744813843210672677, 5.28888958868363812420471932101, 6.84666074039592107350230004766, 7.49674853250841138701759634638, 8.530945589208049100672885995963, 8.656929244624234630824060750928, 9.965088507638647073968732990367

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