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  − 2.29·2-s + 2.37·3-s + 3.27·4-s + 1.23·5-s − 5.45·6-s + 1.58·7-s − 2.91·8-s + 2.65·9-s − 2.82·10-s + 0.886·11-s + 7.77·12-s − 1.28·13-s − 3.64·14-s + 2.93·15-s + 0.154·16-s + 7.29·17-s − 6.09·18-s − 0.683·19-s + 4.03·20-s + 3.77·21-s − 2.03·22-s + 4.28·23-s − 6.93·24-s − 3.48·25-s + 2.95·26-s − 0.819·27-s + 5.19·28-s + ⋯
L(s)  = 1  − 1.62·2-s + 1.37·3-s + 1.63·4-s + 0.551·5-s − 2.22·6-s + 0.600·7-s − 1.03·8-s + 0.885·9-s − 0.894·10-s + 0.267·11-s + 2.24·12-s − 0.356·13-s − 0.974·14-s + 0.756·15-s + 0.0386·16-s + 1.77·17-s − 1.43·18-s − 0.156·19-s + 0.901·20-s + 0.824·21-s − 0.434·22-s + 0.894·23-s − 1.41·24-s − 0.696·25-s + 0.578·26-s − 0.157·27-s + 0.982·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.30918$
$L(\frac12)$  $\approx$  $1.30918$
$L(\frac{3}{2})$   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 \( 1 + 2.29T + 2T^{2} \)
3 \( 1 - 2.37T + 3T^{2} \)
5 \( 1 - 1.23T + 5T^{2} \)
7 \( 1 - 1.58T + 7T^{2} \)
11 \( 1 - 0.886T + 11T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 - 7.29T + 17T^{2} \)
19 \( 1 + 0.683T + 19T^{2} \)
23 \( 1 - 4.28T + 23T^{2} \)
29 \( 1 + 1.56T + 29T^{2} \)
31 \( 1 - 1.05T + 31T^{2} \)
37 \( 1 + 0.285T + 37T^{2} \)
41 \( 1 - 0.174T + 41T^{2} \)
43 \( 1 + 5.95T + 43T^{2} \)
47 \( 1 + 6.98T + 47T^{2} \)
53 \( 1 - 3.12T + 53T^{2} \)
59 \( 1 - 3.82T + 59T^{2} \)
61 \( 1 + 0.632T + 61T^{2} \)
67 \( 1 + 1.10T + 67T^{2} \)
71 \( 1 - 4.39T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 7.13T + 83T^{2} \)
89 \( 1 - 3.05T + 89T^{2} \)
97 \( 1 + 7.38T + 97T^{2} \)
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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

−10.08016440640115432423575425220, −9.623769262544672559322356523087, −8.929726645426276451703564370651, −8.034000347552235617273356921652, −7.72210807014134410891032879796, −6.60875695591857074316583833826, −5.18069523552733700825124582260, −3.50711804556022313240264352321, −2.32905742064293298471702943244, −1.37289052153298972985459062886, 1.37289052153298972985459062886, 2.32905742064293298471702943244, 3.50711804556022313240264352321, 5.18069523552733700825124582260, 6.60875695591857074316583833826, 7.72210807014134410891032879796, 8.034000347552235617273356921652, 8.929726645426276451703564370651, 9.623769262544672559322356523087, 10.08016440640115432423575425220

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