Properties

Label 2-5148-1.1-c1-0-4
Degree $2$
Conductor $5148$
Sign $1$
Analytic cond. $41.1069$
Root an. cond. $6.41147$
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  − 3.22·5-s − 0.874·7-s − 11-s + 13-s + 1.16·17-s − 0.874·19-s − 8.84·23-s + 5.39·25-s − 1.65·29-s − 1.22·31-s + 2.82·35-s − 0.820·37-s − 11.6·41-s + 4.91·43-s + 12.4·47-s − 6.23·49-s − 4.44·53-s + 3.22·55-s + 6.44·59-s + 11.2·61-s − 3.22·65-s − 10.0·67-s + 7.71·71-s + 5.57·73-s + 0.874·77-s + 2.47·79-s − 16.9·83-s + ⋯
L(s)  = 1  − 1.44·5-s − 0.330·7-s − 0.301·11-s + 0.277·13-s + 0.283·17-s − 0.200·19-s − 1.84·23-s + 1.07·25-s − 0.306·29-s − 0.219·31-s + 0.476·35-s − 0.134·37-s − 1.82·41-s + 0.750·43-s + 1.81·47-s − 0.890·49-s − 0.610·53-s + 0.434·55-s + 0.839·59-s + 1.44·61-s − 0.399·65-s − 1.22·67-s + 0.915·71-s + 0.652·73-s + 0.0997·77-s + 0.278·79-s − 1.86·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5148\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(41.1069\)
Root analytic conductor: \(6.41147\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5148,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7916667236\)
\(L(\frac12)\) \(\approx\) \(0.7916667236\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + 3.22T + 5T^{2} \)
7 \( 1 + 0.874T + 7T^{2} \)
17 \( 1 - 1.16T + 17T^{2} \)
19 \( 1 + 0.874T + 19T^{2} \)
23 \( 1 + 8.84T + 23T^{2} \)
29 \( 1 + 1.65T + 29T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 + 0.820T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 - 4.91T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 4.44T + 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 7.71T + 71T^{2} \)
73 \( 1 - 5.57T + 73T^{2} \)
79 \( 1 - 2.47T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−8.255199423954936423032282640205, −7.53513076859194012189054615170, −6.95693451382939580338211120594, −6.06215009275145085073224117597, −5.31395799188060770061743310577, −4.26443754743508329983700495708, −3.82677803960366106719693936237, −3.04539950149323071347593329718, −1.90736625255262595651694519340, −0.46728656984322096029081225310, 0.46728656984322096029081225310, 1.90736625255262595651694519340, 3.04539950149323071347593329718, 3.82677803960366106719693936237, 4.26443754743508329983700495708, 5.31395799188060770061743310577, 6.06215009275145085073224117597, 6.95693451382939580338211120594, 7.53513076859194012189054615170, 8.255199423954936423032282640205

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