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  − 1.10·2-s − 3-s − 0.784·4-s + 0.760·5-s + 1.10·6-s + 7-s + 3.06·8-s + 9-s − 0.838·10-s + 0.597·11-s + 0.784·12-s − 4.75·13-s − 1.10·14-s − 0.760·15-s − 1.81·16-s + 1.90·17-s − 1.10·18-s − 5.87·19-s − 0.596·20-s − 21-s − 0.658·22-s − 7.97·23-s − 3.06·24-s − 4.42·25-s + 5.24·26-s − 27-s − 0.784·28-s + ⋯
L(s)  = 1  − 0.779·2-s − 0.577·3-s − 0.392·4-s + 0.340·5-s + 0.450·6-s + 0.377·7-s + 1.08·8-s + 0.333·9-s − 0.265·10-s + 0.180·11-s + 0.226·12-s − 1.31·13-s − 0.294·14-s − 0.196·15-s − 0.454·16-s + 0.462·17-s − 0.259·18-s − 1.34·19-s − 0.133·20-s − 0.218·21-s − 0.140·22-s − 1.66·23-s − 0.626·24-s − 0.884·25-s + 1.02·26-s − 0.192·27-s − 0.148·28-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  =  $(2,\ 8043,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4759821630$
$L(\frac12)$  $\approx$  $0.4759821630$
$L(\frac{3}{2})$   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 \( 1 + 1.10T + 2T^{2} \)
5 \( 1 - 0.760T + 5T^{2} \)
11 \( 1 - 0.597T + 11T^{2} \)
13 \( 1 + 4.75T + 13T^{2} \)
17 \( 1 - 1.90T + 17T^{2} \)
19 \( 1 + 5.87T + 19T^{2} \)
23 \( 1 + 7.97T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 + 4.54T + 31T^{2} \)
37 \( 1 + 4.13T + 37T^{2} \)
41 \( 1 - 9.73T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 5.17T + 47T^{2} \)
53 \( 1 - 14.0T + 53T^{2} \)
59 \( 1 + 7.09T + 59T^{2} \)
61 \( 1 + 4.13T + 61T^{2} \)
67 \( 1 - 1.80T + 67T^{2} \)
71 \( 1 + 15.7T + 71T^{2} \)
73 \( 1 - 6.46T + 73T^{2} \)
79 \( 1 + 2.06T + 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 - 5.04T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−7.79250236733534017264076251506, −7.40085396913881046367519446467, −6.44760214407993002075851701255, −5.79134690277736270855685245233, −4.98561411797075498134352492014, −4.44140039963535464897443610324, −3.69541183082273199276987635139, −2.22834009277939260312354179018, −1.69274299918259500181182963564, −0.39545914663096894956811092649, 0.39545914663096894956811092649, 1.69274299918259500181182963564, 2.22834009277939260312354179018, 3.69541183082273199276987635139, 4.44140039963535464897443610324, 4.98561411797075498134352492014, 5.79134690277736270855685245233, 6.44760214407993002075851701255, 7.40085396913881046367519446467, 7.79250236733534017264076251506

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