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  + 0.431·2-s − 1.56·3-s − 1.81·4-s + 1.50·5-s − 0.673·6-s − 4.83·7-s − 1.64·8-s − 0.557·9-s + 0.646·10-s + 2.60·11-s + 2.83·12-s + 5.36·13-s − 2.08·14-s − 2.34·15-s + 2.91·16-s + 6.13·17-s − 0.240·18-s + 1.28·19-s − 2.72·20-s + 7.54·21-s + 1.12·22-s − 3.55·23-s + 2.56·24-s − 2.74·25-s + 2.31·26-s + 5.55·27-s + 8.76·28-s + ⋯
L(s)  = 1  + 0.304·2-s − 0.902·3-s − 0.907·4-s + 0.671·5-s − 0.275·6-s − 1.82·7-s − 0.581·8-s − 0.185·9-s + 0.204·10-s + 0.784·11-s + 0.818·12-s + 1.48·13-s − 0.556·14-s − 0.605·15-s + 0.729·16-s + 1.48·17-s − 0.0566·18-s + 0.295·19-s − 0.608·20-s + 1.64·21-s + 0.239·22-s − 0.740·23-s + 0.524·24-s − 0.549·25-s + 0.453·26-s + 1.07·27-s + 1.65·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.910238$
$L(\frac12)$  $\approx$  $0.910238$
$L(\frac{3}{2})$   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 \( 1 - 0.431T + 2T^{2} \)
3 \( 1 + 1.56T + 3T^{2} \)
5 \( 1 - 1.50T + 5T^{2} \)
7 \( 1 + 4.83T + 7T^{2} \)
11 \( 1 - 2.60T + 11T^{2} \)
13 \( 1 - 5.36T + 13T^{2} \)
17 \( 1 - 6.13T + 17T^{2} \)
19 \( 1 - 1.28T + 19T^{2} \)
23 \( 1 + 3.55T + 23T^{2} \)
29 \( 1 + 1.96T + 29T^{2} \)
31 \( 1 - 7.06T + 31T^{2} \)
37 \( 1 - 3.47T + 37T^{2} \)
41 \( 1 - 5.18T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 - 4.95T + 47T^{2} \)
53 \( 1 + 5.85T + 53T^{2} \)
59 \( 1 + 9.60T + 59T^{2} \)
61 \( 1 - 6.86T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 + 5.87T + 71T^{2} \)
73 \( 1 - 4.41T + 73T^{2} \)
79 \( 1 - 8.82T + 79T^{2} \)
83 \( 1 - 1.16T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + 5.05T + 97T^{2} \)
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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

−10.42537839561236483561337554321, −9.732129707095442677966269304801, −9.201160692876829853988364272795, −8.099705677837254906189583282792, −6.38304165732218226228654694726, −6.13297698156052554648254992999, −5.40810628307903611403230035570, −3.92177741916822205069605280969, −3.18972433979458288903329061056, −0.843853602991687097253199822477, 0.843853602991687097253199822477, 3.18972433979458288903329061056, 3.92177741916822205069605280969, 5.40810628307903611403230035570, 6.13297698156052554648254992999, 6.38304165732218226228654694726, 8.099705677837254906189583282792, 9.201160692876829853988364272795, 9.732129707095442677966269304801, 10.42537839561236483561337554321

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