Properties

Degree 2
Conductor $ 7 \cdot 859 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.71·2-s + 3.20·3-s + 5.34·4-s + 3.06·5-s − 8.69·6-s + 7-s − 9.07·8-s + 7.28·9-s − 8.31·10-s − 0.601·11-s + 17.1·12-s − 1.91·13-s − 2.71·14-s + 9.83·15-s + 13.9·16-s + 2.21·17-s − 19.7·18-s − 1.88·19-s + 16.4·20-s + 3.20·21-s + 1.63·22-s + 0.548·23-s − 29.1·24-s + 4.40·25-s + 5.18·26-s + 13.7·27-s + 5.34·28-s + ⋯
L(s)  = 1  − 1.91·2-s + 1.85·3-s + 2.67·4-s + 1.37·5-s − 3.54·6-s + 0.377·7-s − 3.21·8-s + 2.42·9-s − 2.62·10-s − 0.181·11-s + 4.95·12-s − 0.530·13-s − 0.724·14-s + 2.53·15-s + 3.47·16-s + 0.536·17-s − 4.65·18-s − 0.433·19-s + 3.66·20-s + 0.699·21-s + 0.347·22-s + 0.114·23-s − 5.94·24-s + 0.880·25-s + 1.01·26-s + 2.64·27-s + 1.01·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6013\)    =    \(7 \cdot 859\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6013} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6013,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.536351648$
$L(\frac12)$  $\approx$  $2.536351648$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;859\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;859\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 - T \)
859 \( 1 + T \)
good2 \( 1 + 2.71T + 2T^{2} \)
3 \( 1 - 3.20T + 3T^{2} \)
5 \( 1 - 3.06T + 5T^{2} \)
11 \( 1 + 0.601T + 11T^{2} \)
13 \( 1 + 1.91T + 13T^{2} \)
17 \( 1 - 2.21T + 17T^{2} \)
19 \( 1 + 1.88T + 19T^{2} \)
23 \( 1 - 0.548T + 23T^{2} \)
29 \( 1 + 4.40T + 29T^{2} \)
31 \( 1 + 0.578T + 31T^{2} \)
37 \( 1 - 2.37T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 - 9.96T + 43T^{2} \)
47 \( 1 + 2.52T + 47T^{2} \)
53 \( 1 - 1.66T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 4.42T + 61T^{2} \)
67 \( 1 - 4.37T + 67T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 + 9.20T + 79T^{2} \)
83 \( 1 - 4.25T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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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

−8.243452530890466153455962872522, −7.57012884640655197105539813579, −7.26287235446988816935412170919, −6.29319240640697769213256486839, −5.52925663872231183877175250004, −4.14899782705153846471431276934, −2.94787564193931451555445640777, −2.40932328017182096175052572882, −1.89076705361598348755100440647, −1.07921835010577913181629152556, 1.07921835010577913181629152556, 1.89076705361598348755100440647, 2.40932328017182096175052572882, 2.94787564193931451555445640777, 4.14899782705153846471431276934, 5.52925663872231183877175250004, 6.29319240640697769213256486839, 7.26287235446988816935412170919, 7.57012884640655197105539813579, 8.243452530890466153455962872522

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