Properties

Label 2-5796-1.1-c1-0-23
Degree $2$
Conductor $5796$
Sign $1$
Analytic cond. $46.2812$
Root an. cond. $6.80303$
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  + 0.502·5-s + 7-s + 3.41·11-s + 4.73·13-s − 6.89·17-s + 7.51·19-s − 23-s − 4.74·25-s + 0.637·29-s + 5.23·31-s + 0.502·35-s − 5.84·37-s + 5.06·41-s + 4.37·43-s + 11.6·47-s + 49-s − 2.12·53-s + 1.71·55-s + 12.4·59-s − 11.3·61-s + 2.37·65-s − 1.36·67-s + 11.0·71-s + 14.0·73-s + 3.41·77-s − 16.2·79-s − 14.6·83-s + ⋯
L(s)  = 1  + 0.224·5-s + 0.377·7-s + 1.03·11-s + 1.31·13-s − 1.67·17-s + 1.72·19-s − 0.208·23-s − 0.949·25-s + 0.118·29-s + 0.940·31-s + 0.0849·35-s − 0.960·37-s + 0.790·41-s + 0.667·43-s + 1.69·47-s + 0.142·49-s − 0.292·53-s + 0.231·55-s + 1.61·59-s − 1.45·61-s + 0.294·65-s − 0.167·67-s + 1.31·71-s + 1.64·73-s + 0.389·77-s − 1.82·79-s − 1.61·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5796\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(46.2812\)
Root analytic conductor: \(6.80303\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5796,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.582257267\)
\(L(\frac12)\) \(\approx\) \(2.582257267\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good5 \( 1 - 0.502T + 5T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 - 4.73T + 13T^{2} \)
17 \( 1 + 6.89T + 17T^{2} \)
19 \( 1 - 7.51T + 19T^{2} \)
29 \( 1 - 0.637T + 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 - 5.06T + 41T^{2} \)
43 \( 1 - 4.37T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 2.12T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 1.36T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + 4.59T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.203017345548456638582505654313, −7.34240145967444582316841883428, −6.65942821414294969874650008013, −5.99867488330127222876061101264, −5.34022400429200844598937843816, −4.26438661412831363362132292119, −3.85804828535245319911366344252, −2.76885159470983757150324781072, −1.75335054080206791594906327708, −0.911191153798897478834095819448, 0.911191153798897478834095819448, 1.75335054080206791594906327708, 2.76885159470983757150324781072, 3.85804828535245319911366344252, 4.26438661412831363362132292119, 5.34022400429200844598937843816, 5.99867488330127222876061101264, 6.65942821414294969874650008013, 7.34240145967444582316841883428, 8.203017345548456638582505654313

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