Properties

Label 2-7168-1.1-c1-0-158
Degree $2$
Conductor $7168$
Sign $-1$
Analytic cond. $57.2367$
Root an. cond. $7.56549$
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  − 0.463·3-s + 1.98·5-s + 7-s − 2.78·9-s − 0.628·11-s + 3.19·13-s − 0.919·15-s − 4.85·17-s − 0.161·19-s − 0.463·21-s + 3.20·23-s − 1.06·25-s + 2.68·27-s + 1.41·29-s − 5.34·31-s + 0.291·33-s + 1.98·35-s + 2.87·37-s − 1.47·39-s − 9.57·41-s − 9.71·43-s − 5.52·45-s + 9.70·47-s + 49-s + 2.25·51-s − 10.8·53-s − 1.24·55-s + ⋯
L(s)  = 1  − 0.267·3-s + 0.886·5-s + 0.377·7-s − 0.928·9-s − 0.189·11-s + 0.884·13-s − 0.237·15-s − 1.17·17-s − 0.0369·19-s − 0.101·21-s + 0.668·23-s − 0.213·25-s + 0.516·27-s + 0.261·29-s − 0.959·31-s + 0.0507·33-s + 0.335·35-s + 0.473·37-s − 0.236·39-s − 1.49·41-s − 1.48·43-s − 0.823·45-s + 1.41·47-s + 0.142·49-s + 0.315·51-s − 1.48·53-s − 0.167·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7168\)    =    \(2^{10} \cdot 7\)
Sign: $-1$
Analytic conductor: \(57.2367\)
Root analytic conductor: \(7.56549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7168,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + 0.463T + 3T^{2} \)
5 \( 1 - 1.98T + 5T^{2} \)
11 \( 1 + 0.628T + 11T^{2} \)
13 \( 1 - 3.19T + 13T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 + 0.161T + 19T^{2} \)
23 \( 1 - 3.20T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 5.34T + 31T^{2} \)
37 \( 1 - 2.87T + 37T^{2} \)
41 \( 1 + 9.57T + 41T^{2} \)
43 \( 1 + 9.71T + 43T^{2} \)
47 \( 1 - 9.70T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 2.12T + 59T^{2} \)
61 \( 1 - 2.46T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 - 7.18T + 71T^{2} \)
73 \( 1 + 9.04T + 73T^{2} \)
79 \( 1 - 9.58T + 79T^{2} \)
83 \( 1 - 7.49T + 83T^{2} \)
89 \( 1 + 2.49T + 89T^{2} \)
97 \( 1 - 5.89T + 97T^{2} \)
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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

−7.61255253816115305725897486922, −6.58844167244017521629874426595, −6.27140035815116886724910000080, −5.40978725295149996281119659958, −4.99381979347095794692409617637, −3.96945984845548063450602905758, −3.05270681548811774696162067525, −2.20347332229359729963753879216, −1.39203610231997976625397438559, 0, 1.39203610231997976625397438559, 2.20347332229359729963753879216, 3.05270681548811774696162067525, 3.96945984845548063450602905758, 4.99381979347095794692409617637, 5.40978725295149996281119659958, 6.27140035815116886724910000080, 6.58844167244017521629874426595, 7.61255253816115305725897486922

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