Properties

Label 2-8043-1.1-c1-0-49
Degree $2$
Conductor $8043$
Sign $1$
Analytic cond. $64.2236$
Root an. cond. $8.01396$
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  − 2.63·2-s − 3-s + 4.92·4-s + 3.86·5-s + 2.63·6-s − 7-s − 7.70·8-s + 9-s − 10.1·10-s − 3.94·11-s − 4.92·12-s + 0.666·13-s + 2.63·14-s − 3.86·15-s + 10.4·16-s − 4.28·17-s − 2.63·18-s + 0.176·19-s + 19.0·20-s + 21-s + 10.3·22-s − 5.36·23-s + 7.70·24-s + 9.94·25-s − 1.75·26-s − 27-s − 4.92·28-s + ⋯
L(s)  = 1  − 1.86·2-s − 0.577·3-s + 2.46·4-s + 1.72·5-s + 1.07·6-s − 0.377·7-s − 2.72·8-s + 0.333·9-s − 3.21·10-s − 1.19·11-s − 1.42·12-s + 0.184·13-s + 0.703·14-s − 0.998·15-s + 2.60·16-s − 1.03·17-s − 0.620·18-s + 0.0404·19-s + 4.26·20-s + 0.218·21-s + 2.21·22-s − 1.11·23-s + 1.57·24-s + 1.98·25-s − 0.344·26-s − 0.192·27-s − 0.931·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8043\)    =    \(3 \cdot 7 \cdot 383\)
Sign: $1$
Analytic conductor: \(64.2236\)
Root analytic conductor: \(8.01396\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8043,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6095113803\)
\(L(\frac12)\) \(\approx\) \(0.6095113803\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
383 \( 1 - T \)
good2 \( 1 + 2.63T + 2T^{2} \)
5 \( 1 - 3.86T + 5T^{2} \)
11 \( 1 + 3.94T + 11T^{2} \)
13 \( 1 - 0.666T + 13T^{2} \)
17 \( 1 + 4.28T + 17T^{2} \)
19 \( 1 - 0.176T + 19T^{2} \)
23 \( 1 + 5.36T + 23T^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 + 7.79T + 31T^{2} \)
37 \( 1 - 9.69T + 37T^{2} \)
41 \( 1 - 6.28T + 41T^{2} \)
43 \( 1 - 3.37T + 43T^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 + 5.03T + 53T^{2} \)
59 \( 1 + 4.21T + 59T^{2} \)
61 \( 1 + 6.76T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 3.54T + 71T^{2} \)
73 \( 1 - 1.67T + 73T^{2} \)
79 \( 1 + 5.00T + 79T^{2} \)
83 \( 1 + 2.95T + 83T^{2} \)
89 \( 1 - 4.67T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−7.74606171361924840475495481943, −7.36353835057410123525773179419, −6.46149851627029677778784345418, −5.90778617223245829989087231481, −5.64204719071216312726194760384, −4.39840418022804187995077550269, −2.87270793070850968039902776253, −2.24955195958961981762962406430, −1.66030178814629708991156140276, −0.51483249711951446412569252496, 0.51483249711951446412569252496, 1.66030178814629708991156140276, 2.24955195958961981762962406430, 2.87270793070850968039902776253, 4.39840418022804187995077550269, 5.64204719071216312726194760384, 5.90778617223245829989087231481, 6.46149851627029677778784345418, 7.36353835057410123525773179419, 7.74606171361924840475495481943

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