Properties

Degree 2
Conductor $ 13 \cdot 619 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.51·2-s − 2.93·3-s + 4.30·4-s − 3.06·5-s + 7.36·6-s + 3.82·7-s − 5.78·8-s + 5.60·9-s + 7.70·10-s − 0.749·11-s − 12.6·12-s + 13-s − 9.59·14-s + 8.99·15-s + 5.92·16-s − 6.57·17-s − 14.0·18-s + 5.95·19-s − 13.2·20-s − 11.2·21-s + 1.88·22-s + 7.17·23-s + 16.9·24-s + 4.40·25-s − 2.51·26-s − 7.62·27-s + 16.4·28-s + ⋯
L(s)  = 1  − 1.77·2-s − 1.69·3-s + 2.15·4-s − 1.37·5-s + 3.00·6-s + 1.44·7-s − 2.04·8-s + 1.86·9-s + 2.43·10-s − 0.225·11-s − 3.64·12-s + 0.277·13-s − 2.56·14-s + 2.32·15-s + 1.48·16-s − 1.59·17-s − 3.31·18-s + 1.36·19-s − 2.95·20-s − 2.44·21-s + 0.401·22-s + 1.49·23-s + 3.46·24-s + 0.881·25-s − 0.492·26-s − 1.46·27-s + 3.10·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8047\)    =    \(13 \cdot 619\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8047} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8047,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{13,\;619\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{13,\;619\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 - T \)
619 \( 1 - T \)
good2 \( 1 + 2.51T + 2T^{2} \)
3 \( 1 + 2.93T + 3T^{2} \)
5 \( 1 + 3.06T + 5T^{2} \)
7 \( 1 - 3.82T + 7T^{2} \)
11 \( 1 + 0.749T + 11T^{2} \)
17 \( 1 + 6.57T + 17T^{2} \)
19 \( 1 - 5.95T + 19T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 - 0.412T + 29T^{2} \)
31 \( 1 - 0.475T + 31T^{2} \)
37 \( 1 - 7.83T + 37T^{2} \)
41 \( 1 + 6.09T + 41T^{2} \)
43 \( 1 - 3.56T + 43T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + 8.83T + 59T^{2} \)
61 \( 1 + 4.24T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 2.64T + 73T^{2} \)
79 \( 1 + 4.45T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 - 18.6T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−7.64436933707594159799756956909, −6.97530099303368132414359068214, −6.50047352627592148014100597240, −5.41692297892231750209921767206, −4.80248557246489138636491313579, −4.15559133479275932930834464640, −2.78420392758345624440255538831, −1.51364025674112298102868766016, −0.895514942517053520897768227950, 0, 0.895514942517053520897768227950, 1.51364025674112298102868766016, 2.78420392758345624440255538831, 4.15559133479275932930834464640, 4.80248557246489138636491313579, 5.41692297892231750209921767206, 6.50047352627592148014100597240, 6.97530099303368132414359068214, 7.64436933707594159799756956909

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