Properties

Label 2-5148-1.1-c1-0-27
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  + 3.41·5-s + 2.82·7-s + 11-s − 13-s + 6.24·17-s + 4.82·23-s + 6.65·25-s + 0.585·29-s − 5.41·31-s + 9.65·35-s + 0.828·37-s − 3.65·41-s − 0.242·43-s + 8·47-s + 1.00·49-s − 9.31·53-s + 3.41·55-s − 2.82·59-s − 3.17·61-s − 3.41·65-s + 4.24·67-s + 10.8·71-s − 5.65·73-s + 2.82·77-s − 7.07·79-s + 17.6·83-s + 21.3·85-s + ⋯
L(s)  = 1  + 1.52·5-s + 1.06·7-s + 0.301·11-s − 0.277·13-s + 1.51·17-s + 1.00·23-s + 1.33·25-s + 0.108·29-s − 0.972·31-s + 1.63·35-s + 0.136·37-s − 0.571·41-s − 0.0370·43-s + 1.16·47-s + 0.142·49-s − 1.27·53-s + 0.460·55-s − 0.368·59-s − 0.406·61-s − 0.423·65-s + 0.518·67-s + 1.28·71-s − 0.662·73-s + 0.322·77-s − 0.795·79-s + 1.93·83-s + 2.31·85-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

\(L(1)\) \(\approx\) \(3.459006239\)
\(L(\frac12)\) \(\approx\) \(3.459006239\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 5T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
17 \( 1 - 6.24T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 - 0.585T + 29T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 - 0.828T + 37T^{2} \)
41 \( 1 + 3.65T + 41T^{2} \)
43 \( 1 + 0.242T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 9.31T + 53T^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 3.17T + 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 5.65T + 73T^{2} \)
79 \( 1 + 7.07T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−8.164865654332625290356180354060, −7.55528092228218524682115413905, −6.71961419306795465665902554781, −5.94083768853498501191655027651, −5.26691948657098810447211358853, −4.86802388524843408253062910108, −3.65487691527487876452450849634, −2.68872955176590675630986868630, −1.77566351478311252334918015151, −1.13901250662942083927412051578, 1.13901250662942083927412051578, 1.77566351478311252334918015151, 2.68872955176590675630986868630, 3.65487691527487876452450849634, 4.86802388524843408253062910108, 5.26691948657098810447211358853, 5.94083768853498501191655027651, 6.71961419306795465665902554781, 7.55528092228218524682115413905, 8.164865654332625290356180354060

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