Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 383 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.890·2-s − 3-s − 1.20·4-s − 2.94·5-s + 0.890·6-s + 7-s + 2.85·8-s + 9-s + 2.62·10-s + 0.179·11-s + 1.20·12-s − 1.10·13-s − 0.890·14-s + 2.94·15-s − 0.128·16-s + 5.29·17-s − 0.890·18-s + 7.81·19-s + 3.55·20-s − 21-s − 0.159·22-s − 0.108·23-s − 2.85·24-s + 3.67·25-s + 0.984·26-s − 27-s − 1.20·28-s + ⋯
L(s)  = 1  − 0.629·2-s − 0.577·3-s − 0.603·4-s − 1.31·5-s + 0.363·6-s + 0.377·7-s + 1.00·8-s + 0.333·9-s + 0.829·10-s + 0.0541·11-s + 0.348·12-s − 0.306·13-s − 0.237·14-s + 0.760·15-s − 0.0321·16-s + 1.28·17-s − 0.209·18-s + 1.79·19-s + 0.795·20-s − 0.218·21-s − 0.0340·22-s − 0.0226·23-s − 0.582·24-s + 0.735·25-s + 0.192·26-s − 0.192·27-s − 0.228·28-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

\( d \)  =  \(2\)
\( N \)  =  \(8043\)    =    \(3 \cdot 7 \cdot 383\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8043} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8043,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9855454759$
$L(\frac12)$  $\approx$  $0.9855454759$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;383\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;383\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 - T \)
383 \( 1 + T \)
good2 \( 1 + 0.890T + 2T^{2} \)
5 \( 1 + 2.94T + 5T^{2} \)
11 \( 1 - 0.179T + 11T^{2} \)
13 \( 1 + 1.10T + 13T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 7.81T + 19T^{2} \)
23 \( 1 + 0.108T + 23T^{2} \)
29 \( 1 - 8.19T + 29T^{2} \)
31 \( 1 - 7.92T + 31T^{2} \)
37 \( 1 - 6.33T + 37T^{2} \)
41 \( 1 - 9.17T + 41T^{2} \)
43 \( 1 + 3.90T + 43T^{2} \)
47 \( 1 - 5.06T + 47T^{2} \)
53 \( 1 - 3.78T + 53T^{2} \)
59 \( 1 + 2.54T + 59T^{2} \)
61 \( 1 + 3.05T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 4.82T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 8.14T + 79T^{2} \)
83 \( 1 + 2.18T + 83T^{2} \)
89 \( 1 + 18.0T + 89T^{2} \)
97 \( 1 - 3.58T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.87735150076813537649814272258, −7.48321082161164505204542118317, −6.64050534031483975357011365204, −5.57560753544359954663435264812, −4.97813006924827110775438705602, −4.35157388029554068134776923740, −3.67177419304119246669850341668, −2.75316101075224671249083332052, −1.07688591919141700502708661107, −0.75322040111704202158453596170, 0.75322040111704202158453596170, 1.07688591919141700502708661107, 2.75316101075224671249083332052, 3.67177419304119246669850341668, 4.35157388029554068134776923740, 4.97813006924827110775438705602, 5.57560753544359954663435264812, 6.64050534031483975357011365204, 7.48321082161164505204542118317, 7.87735150076813537649814272258

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