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  − 1.90·2-s − 3-s + 1.61·4-s + 0.767·5-s + 1.90·6-s + 7-s + 0.727·8-s + 9-s − 1.45·10-s + 0.295·11-s − 1.61·12-s + 3.77·13-s − 1.90·14-s − 0.767·15-s − 4.61·16-s + 5.24·17-s − 1.90·18-s − 3.76·19-s + 1.24·20-s − 21-s − 0.561·22-s − 1.71·23-s − 0.727·24-s − 4.41·25-s − 7.18·26-s − 27-s + 1.61·28-s + ⋯
L(s)  = 1  − 1.34·2-s − 0.577·3-s + 0.808·4-s + 0.343·5-s + 0.776·6-s + 0.377·7-s + 0.257·8-s + 0.333·9-s − 0.461·10-s + 0.0890·11-s − 0.466·12-s + 1.04·13-s − 0.508·14-s − 0.198·15-s − 1.15·16-s + 1.27·17-s − 0.448·18-s − 0.863·19-s + 0.277·20-s − 0.218·21-s − 0.119·22-s − 0.357·23-s − 0.148·24-s − 0.882·25-s − 1.40·26-s − 0.192·27-s + 0.305·28-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  =  $(2,\ 8043,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8330071895$
$L(\frac12)$  $\approx$  $0.8330071895$
$L(\frac{3}{2})$   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 \( 1 + 1.90T + 2T^{2} \)
5 \( 1 - 0.767T + 5T^{2} \)
11 \( 1 - 0.295T + 11T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 + 3.76T + 19T^{2} \)
23 \( 1 + 1.71T + 23T^{2} \)
29 \( 1 + 7.83T + 29T^{2} \)
31 \( 1 - 0.600T + 31T^{2} \)
37 \( 1 + 3.78T + 37T^{2} \)
41 \( 1 - 8.68T + 41T^{2} \)
43 \( 1 - 2.68T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 7.17T + 53T^{2} \)
59 \( 1 - 9.38T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 - 0.510T + 67T^{2} \)
71 \( 1 - 7.64T + 71T^{2} \)
73 \( 1 - 6.16T + 73T^{2} \)
79 \( 1 + 1.92T + 79T^{2} \)
83 \( 1 + 4.65T + 83T^{2} \)
89 \( 1 - 5.88T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−8.044376290142877330533506277333, −7.32714977453265002204352056100, −6.54983244579723952254832602450, −5.86765813837284593838039281743, −5.26181414437402770059730308167, −4.24277640210099935960226025849, −3.56417572962776201760734105814, −2.13550177497835146744496736780, −1.52838587573531509261269008382, −0.60892641844887014800892847751, 0.60892641844887014800892847751, 1.52838587573531509261269008382, 2.13550177497835146744496736780, 3.56417572962776201760734105814, 4.24277640210099935960226025849, 5.26181414437402770059730308167, 5.86765813837284593838039281743, 6.54983244579723952254832602450, 7.32714977453265002204352056100, 8.044376290142877330533506277333

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