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  − 1.98·2-s − 1.72·3-s + 1.93·4-s − 1.05·5-s + 3.41·6-s + 3.75·7-s + 0.132·8-s − 0.0271·9-s + 2.09·10-s + 6.08·11-s − 3.33·12-s − 6.72·13-s − 7.44·14-s + 1.82·15-s − 4.12·16-s − 5.23·17-s + 0.0539·18-s − 0.354·19-s − 2.04·20-s − 6.47·21-s − 12.0·22-s + 6.00·23-s − 0.228·24-s − 3.88·25-s + 13.3·26-s + 5.21·27-s + 7.25·28-s + ⋯
L(s)  = 1  − 1.40·2-s − 0.995·3-s + 0.966·4-s − 0.473·5-s + 1.39·6-s + 1.41·7-s + 0.0468·8-s − 0.00906·9-s + 0.663·10-s + 1.83·11-s − 0.962·12-s − 1.86·13-s − 1.99·14-s + 0.470·15-s − 1.03·16-s − 1.26·17-s + 0.0127·18-s − 0.0814·19-s − 0.457·20-s − 1.41·21-s − 2.57·22-s + 1.25·23-s − 0.0465·24-s − 0.776·25-s + 2.61·26-s + 1.00·27-s + 1.37·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.476053$
$L(\frac12)$  $\approx$  $0.476053$
$L(\frac{3}{2})$   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 \( 1 + 1.98T + 2T^{2} \)
3 \( 1 + 1.72T + 3T^{2} \)
5 \( 1 + 1.05T + 5T^{2} \)
7 \( 1 - 3.75T + 7T^{2} \)
11 \( 1 - 6.08T + 11T^{2} \)
13 \( 1 + 6.72T + 13T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 + 0.354T + 19T^{2} \)
23 \( 1 - 6.00T + 23T^{2} \)
29 \( 1 - 6.90T + 29T^{2} \)
31 \( 1 + 5.33T + 31T^{2} \)
37 \( 1 - 5.05T + 37T^{2} \)
41 \( 1 - 9.63T + 41T^{2} \)
43 \( 1 + 0.0907T + 43T^{2} \)
47 \( 1 + 4.97T + 47T^{2} \)
53 \( 1 + 6.61T + 53T^{2} \)
59 \( 1 - 6.12T + 59T^{2} \)
61 \( 1 - 1.72T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 5.79T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 0.184T + 83T^{2} \)
89 \( 1 + 7.11T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−10.76788320266700613909558874017, −9.607327195846374853034741716783, −8.954412755970775217488944490111, −8.078547700229649108668016673661, −7.20540311788047990695968038180, −6.49324912056451911790463182677, −4.96289152021834544722611074925, −4.39179835950707295807812093707, −2.13512872343296515296372422759, −0.78666677260535171729752320000, 0.78666677260535171729752320000, 2.13512872343296515296372422759, 4.39179835950707295807812093707, 4.96289152021834544722611074925, 6.49324912056451911790463182677, 7.20540311788047990695968038180, 8.078547700229649108668016673661, 8.954412755970775217488944490111, 9.607327195846374853034741716783, 10.76788320266700613909558874017

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