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  − 2.64·2-s + 0.320·3-s + 4.98·4-s + 3.17·5-s − 0.847·6-s + 7-s − 7.87·8-s − 2.89·9-s − 8.39·10-s − 2.25·11-s + 1.59·12-s − 4.14·13-s − 2.64·14-s + 1.01·15-s + 10.8·16-s − 4.34·17-s + 7.65·18-s − 3.45·19-s + 15.8·20-s + 0.320·21-s + 5.95·22-s − 0.0874·23-s − 2.52·24-s + 5.08·25-s + 10.9·26-s − 1.89·27-s + 4.98·28-s + ⋯
L(s)  = 1  − 1.86·2-s + 0.185·3-s + 2.49·4-s + 1.42·5-s − 0.345·6-s + 0.377·7-s − 2.78·8-s − 0.965·9-s − 2.65·10-s − 0.679·11-s + 0.461·12-s − 1.14·13-s − 0.706·14-s + 0.262·15-s + 2.71·16-s − 1.05·17-s + 1.80·18-s − 0.793·19-s + 3.53·20-s + 0.0699·21-s + 1.27·22-s − 0.0182·23-s − 0.515·24-s + 1.01·25-s + 2.14·26-s − 0.363·27-s + 0.941·28-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  =  $(2,\ 6013,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7683889091$
$L(\frac12)$  $\approx$  $0.7683889091$
$L(\frac{3}{2})$   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 \( 1 + 2.64T + 2T^{2} \)
3 \( 1 - 0.320T + 3T^{2} \)
5 \( 1 - 3.17T + 5T^{2} \)
11 \( 1 + 2.25T + 11T^{2} \)
13 \( 1 + 4.14T + 13T^{2} \)
17 \( 1 + 4.34T + 17T^{2} \)
19 \( 1 + 3.45T + 19T^{2} \)
23 \( 1 + 0.0874T + 23T^{2} \)
29 \( 1 - 8.08T + 29T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 - 0.513T + 41T^{2} \)
43 \( 1 - 1.83T + 43T^{2} \)
47 \( 1 - 3.62T + 47T^{2} \)
53 \( 1 - 0.723T + 53T^{2} \)
59 \( 1 + 6.53T + 59T^{2} \)
61 \( 1 + 0.324T + 61T^{2} \)
67 \( 1 - 7.47T + 67T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 + 1.38T + 73T^{2} \)
79 \( 1 - 4.31T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 - 0.168T + 89T^{2} \)
97 \( 1 - 4.41T + 97T^{2} \)
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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

−8.208219135136493769052475607700, −7.67372293356177796766300742399, −6.75623889448714671996487001333, −6.25777107482994539548401474682, −5.51359773722449935887743975718, −4.65697444976682764975564772261, −2.90055988073857512351126630446, −2.35288333857932920022370422603, −1.91661664426860877054682893235, −0.57019783703156924373387695143, 0.57019783703156924373387695143, 1.91661664426860877054682893235, 2.35288333857932920022370422603, 2.90055988073857512351126630446, 4.65697444976682764975564772261, 5.51359773722449935887743975718, 6.25777107482994539548401474682, 6.75623889448714671996487001333, 7.67372293356177796766300742399, 8.208219135136493769052475607700

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