Properties

Degree 2
Conductor $ 13 \cdot 463 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.26·2-s + 2.92·3-s + 3.12·4-s + 0.680·5-s − 6.63·6-s − 3.22·7-s − 2.54·8-s + 5.58·9-s − 1.53·10-s + 3.31·11-s + 9.15·12-s − 13-s + 7.29·14-s + 1.99·15-s − 0.491·16-s − 1.52·17-s − 12.6·18-s + 1.11·19-s + 2.12·20-s − 9.44·21-s − 7.49·22-s + 3.23·23-s − 7.44·24-s − 4.53·25-s + 2.26·26-s + 7.57·27-s − 10.0·28-s + ⋯
L(s)  = 1  − 1.60·2-s + 1.69·3-s + 1.56·4-s + 0.304·5-s − 2.70·6-s − 1.21·7-s − 0.898·8-s + 1.86·9-s − 0.486·10-s + 0.998·11-s + 2.64·12-s − 0.277·13-s + 1.94·14-s + 0.514·15-s − 0.122·16-s − 0.369·17-s − 2.97·18-s + 0.255·19-s + 0.475·20-s − 2.05·21-s − 1.59·22-s + 0.673·23-s − 1.52·24-s − 0.907·25-s + 0.443·26-s + 1.45·27-s − 1.90·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6019\)    =    \(13 \cdot 463\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6019} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 6019,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{13,\;463\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{13,\;463\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + T \)
463 \( 1 + T \)
good2 \( 1 + 2.26T + 2T^{2} \)
3 \( 1 - 2.92T + 3T^{2} \)
5 \( 1 - 0.680T + 5T^{2} \)
7 \( 1 + 3.22T + 7T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
17 \( 1 + 1.52T + 17T^{2} \)
19 \( 1 - 1.11T + 19T^{2} \)
23 \( 1 - 3.23T + 23T^{2} \)
29 \( 1 - 0.0198T + 29T^{2} \)
31 \( 1 + 7.73T + 31T^{2} \)
37 \( 1 + 9.89T + 37T^{2} \)
41 \( 1 + 5.18T + 41T^{2} \)
43 \( 1 - 3.84T + 43T^{2} \)
47 \( 1 + 9.40T + 47T^{2} \)
53 \( 1 + 6.20T + 53T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 7.88T + 67T^{2} \)
71 \( 1 + 4.66T + 71T^{2} \)
73 \( 1 - 2.12T + 73T^{2} \)
79 \( 1 + 7.87T + 79T^{2} \)
83 \( 1 - 5.81T + 83T^{2} \)
89 \( 1 + 5.39T + 89T^{2} \)
97 \( 1 - 4.09T + 97T^{2} \)
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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

−7.893946635446765324856142083396, −7.22491630230203234176383394829, −6.82273872202189474978847614481, −6.01154796515955289400953539590, −4.60181429262375772368935865514, −3.51610544421459373595196060072, −3.11321345624189692988220587921, −2.03095630502001814283334099936, −1.51986004578180651311676032989, 0, 1.51986004578180651311676032989, 2.03095630502001814283334099936, 3.11321345624189692988220587921, 3.51610544421459373595196060072, 4.60181429262375772368935865514, 6.01154796515955289400953539590, 6.82273872202189474978847614481, 7.22491630230203234176383394829, 7.893946635446765324856142083396

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