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  − 1.26·2-s + 3.28·3-s − 0.388·4-s + 2.23·5-s − 4.17·6-s + 3.15·7-s + 3.03·8-s + 7.80·9-s − 2.83·10-s + 1.25·11-s − 1.27·12-s + 0.135·13-s − 4.00·14-s + 7.34·15-s − 3.07·16-s − 8.12·17-s − 9.90·18-s − 6.16·19-s − 0.869·20-s + 10.3·21-s − 1.59·22-s − 6.32·23-s + 9.96·24-s − 0.00254·25-s − 0.172·26-s + 15.7·27-s − 1.22·28-s + ⋯
L(s)  = 1  − 0.897·2-s + 1.89·3-s − 0.194·4-s + 0.999·5-s − 1.70·6-s + 1.19·7-s + 1.07·8-s + 2.60·9-s − 0.897·10-s + 0.378·11-s − 0.369·12-s + 0.0376·13-s − 1.06·14-s + 1.89·15-s − 0.767·16-s − 1.97·17-s − 2.33·18-s − 1.41·19-s − 0.194·20-s + 2.26·21-s − 0.339·22-s − 1.31·23-s + 2.03·24-s − 0.000509·25-s − 0.0337·26-s + 3.03·27-s − 0.231·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.05729$
$L(\frac12)$  $\approx$  $2.05729$
$L(\frac{3}{2})$   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 \( 1 + 1.26T + 2T^{2} \)
3 \( 1 - 3.28T + 3T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
7 \( 1 - 3.15T + 7T^{2} \)
11 \( 1 - 1.25T + 11T^{2} \)
13 \( 1 - 0.135T + 13T^{2} \)
17 \( 1 + 8.12T + 17T^{2} \)
19 \( 1 + 6.16T + 19T^{2} \)
23 \( 1 + 6.32T + 23T^{2} \)
29 \( 1 - 4.55T + 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 + 6.68T + 37T^{2} \)
41 \( 1 - 2.34T + 41T^{2} \)
43 \( 1 - 8.30T + 43T^{2} \)
47 \( 1 + 1.40T + 47T^{2} \)
53 \( 1 + 6.60T + 53T^{2} \)
59 \( 1 - 1.32T + 59T^{2} \)
61 \( 1 + 0.790T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 3.99T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 - 3.82T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 3.94T + 89T^{2} \)
97 \( 1 + 1.33T + 97T^{2} \)
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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

−10.31267122868205272446385112493, −9.362251649607263615945472676547, −8.876412216699152717861158198433, −8.334723114874212027495087634497, −7.57285870145561503082029340158, −6.47767927838726558072037285826, −4.68311997632134368178318646214, −4.02560500028598948480984417894, −2.10147256130631049724236624645, −1.85280751890730749370709431071, 1.85280751890730749370709431071, 2.10147256130631049724236624645, 4.02560500028598948480984417894, 4.68311997632134368178318646214, 6.47767927838726558072037285826, 7.57285870145561503082029340158, 8.334723114874212027495087634497, 8.876412216699152717861158198433, 9.362251649607263615945472676547, 10.31267122868205272446385112493

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