Properties

Label 2-7168-1.1-c1-0-144
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  + 0.242·3-s + 0.379·5-s + 7-s − 2.94·9-s − 2.61·11-s − 2.31·13-s + 0.0919·15-s + 7.37·17-s − 5.44·19-s + 0.242·21-s + 6.44·23-s − 4.85·25-s − 1.43·27-s + 5.07·29-s + 6.10·31-s − 0.633·33-s + 0.379·35-s − 10.4·37-s − 0.560·39-s − 0.836·41-s + 5.50·43-s − 1.11·45-s − 6.02·47-s + 49-s + 1.78·51-s + 0.813·53-s − 0.992·55-s + ⋯
L(s)  = 1  + 0.139·3-s + 0.169·5-s + 0.377·7-s − 0.980·9-s − 0.788·11-s − 0.641·13-s + 0.0237·15-s + 1.78·17-s − 1.24·19-s + 0.0528·21-s + 1.34·23-s − 0.971·25-s − 0.276·27-s + 0.941·29-s + 1.09·31-s − 0.110·33-s + 0.0641·35-s − 1.72·37-s − 0.0896·39-s − 0.130·41-s + 0.838·43-s − 0.166·45-s − 0.878·47-s + 0.142·49-s + 0.249·51-s + 0.111·53-s − 0.133·55-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: $\chi_{7168} (1, \cdot )$
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 \( 1 - 0.242T + 3T^{2} \)
5 \( 1 - 0.379T + 5T^{2} \)
11 \( 1 + 2.61T + 11T^{2} \)
13 \( 1 + 2.31T + 13T^{2} \)
17 \( 1 - 7.37T + 17T^{2} \)
19 \( 1 + 5.44T + 19T^{2} \)
23 \( 1 - 6.44T + 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 - 6.10T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 + 0.836T + 41T^{2} \)
43 \( 1 - 5.50T + 43T^{2} \)
47 \( 1 + 6.02T + 47T^{2} \)
53 \( 1 - 0.813T + 53T^{2} \)
59 \( 1 - 7.53T + 59T^{2} \)
61 \( 1 - 1.31T + 61T^{2} \)
67 \( 1 - 8.79T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 3.68T + 73T^{2} \)
79 \( 1 + 4.21T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 - 9.32T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−7.68522828381668310523418378774, −6.94879801367451349205663923933, −6.09091639879590567177830377132, −5.35081180679944895688757727024, −4.96903824786818195094012470461, −3.90856659982161983655468260835, −2.94947927851033178381136746650, −2.47184037466020622321820481914, −1.29723936560942170952227275283, 0, 1.29723936560942170952227275283, 2.47184037466020622321820481914, 2.94947927851033178381136746650, 3.90856659982161983655468260835, 4.96903824786818195094012470461, 5.35081180679944895688757727024, 6.09091639879590567177830377132, 6.94879801367451349205663923933, 7.68522828381668310523418378774

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