Properties

Degree 2
Conductor 619
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.192·2-s − 3.10·3-s − 1.96·4-s + 2.33·5-s + 0.599·6-s − 3.05·7-s + 0.764·8-s + 6.67·9-s − 0.450·10-s − 6.12·11-s + 6.10·12-s − 5.12·13-s + 0.588·14-s − 7.25·15-s + 3.77·16-s − 0.896·17-s − 1.28·18-s + 2.17·19-s − 4.58·20-s + 9.49·21-s + 1.18·22-s + 4.34·23-s − 2.37·24-s + 0.447·25-s + 0.987·26-s − 11.4·27-s + 5.99·28-s + ⋯
L(s)  = 1  − 0.136·2-s − 1.79·3-s − 0.981·4-s + 1.04·5-s + 0.244·6-s − 1.15·7-s + 0.270·8-s + 2.22·9-s − 0.142·10-s − 1.84·11-s + 1.76·12-s − 1.42·13-s + 0.157·14-s − 1.87·15-s + 0.944·16-s − 0.217·17-s − 0.303·18-s + 0.498·19-s − 1.02·20-s + 2.07·21-s + 0.251·22-s + 0.906·23-s − 0.485·24-s + 0.0895·25-s + 0.193·26-s − 2.19·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(619\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{619} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.378849$
$L(\frac12)$  $\approx$  $0.378849$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 619$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 619$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad619 \( 1 - T \)
good2 \( 1 + 0.192T + 2T^{2} \)
3 \( 1 + 3.10T + 3T^{2} \)
5 \( 1 - 2.33T + 5T^{2} \)
7 \( 1 + 3.05T + 7T^{2} \)
11 \( 1 + 6.12T + 11T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 + 0.896T + 17T^{2} \)
19 \( 1 - 2.17T + 19T^{2} \)
23 \( 1 - 4.34T + 23T^{2} \)
29 \( 1 - 3.04T + 29T^{2} \)
31 \( 1 - 2.83T + 31T^{2} \)
37 \( 1 - 6.95T + 37T^{2} \)
41 \( 1 - 3.82T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 + 2.38T + 53T^{2} \)
59 \( 1 - 7.29T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 2.68T + 67T^{2} \)
71 \( 1 - 6.53T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 9.47T + 79T^{2} \)
83 \( 1 - 2.39T + 83T^{2} \)
89 \( 1 + 5.93T + 89T^{2} \)
97 \( 1 - 9.10T + 97T^{2} \)
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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

−10.28316725104283496825570898467, −9.984393539984904882949615883417, −9.397531088705177933916927313840, −7.79800072979111049956940451627, −6.83432856868843765353608767398, −5.80971757050473988773137625489, −5.27875992065282003071978756662, −4.56296173391895580259560170246, −2.68675408720332401052639670935, −0.57424308568506499240605936784, 0.57424308568506499240605936784, 2.68675408720332401052639670935, 4.56296173391895580259560170246, 5.27875992065282003071978756662, 5.80971757050473988773137625489, 6.83432856868843765353608767398, 7.79800072979111049956940451627, 9.397531088705177933916927313840, 9.984393539984904882949615883417, 10.28316725104283496825570898467

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