Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.45·2-s + 2.41·3-s + 0.117·4-s − 2.25·5-s − 3.50·6-s − 1.27·7-s + 2.73·8-s + 2.80·9-s + 3.28·10-s + 2.81·11-s + 0.282·12-s + 1.19·13-s + 1.84·14-s − 5.43·15-s − 4.22·16-s − 3.27·17-s − 4.08·18-s + 4.87·19-s − 0.264·20-s − 3.06·21-s − 4.10·22-s + 9.08·23-s + 6.60·24-s + 0.0853·25-s − 1.74·26-s − 0.458·27-s − 0.149·28-s + ⋯
L(s)  = 1  − 1.02·2-s + 1.39·3-s + 0.0586·4-s − 1.00·5-s − 1.43·6-s − 0.480·7-s + 0.968·8-s + 0.936·9-s + 1.03·10-s + 0.850·11-s + 0.0816·12-s + 0.332·13-s + 0.494·14-s − 1.40·15-s − 1.05·16-s − 0.795·17-s − 0.963·18-s + 1.11·19-s − 0.0591·20-s − 0.668·21-s − 0.874·22-s + 1.89·23-s + 1.34·24-s + 0.0170·25-s − 0.342·26-s − 0.0882·27-s − 0.0281·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.10240$
$L(\frac12)$  $\approx$  $1.10240$
$L(\frac{3}{2})$   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 \( 1 + 1.45T + 2T^{2} \)
3 \( 1 - 2.41T + 3T^{2} \)
5 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 + 1.27T + 7T^{2} \)
11 \( 1 - 2.81T + 11T^{2} \)
13 \( 1 - 1.19T + 13T^{2} \)
17 \( 1 + 3.27T + 17T^{2} \)
19 \( 1 - 4.87T + 19T^{2} \)
23 \( 1 - 9.08T + 23T^{2} \)
29 \( 1 - 8.59T + 29T^{2} \)
31 \( 1 - 9.78T + 31T^{2} \)
37 \( 1 + 5.98T + 37T^{2} \)
41 \( 1 + 0.101T + 41T^{2} \)
43 \( 1 - 0.656T + 43T^{2} \)
47 \( 1 - 8.22T + 47T^{2} \)
53 \( 1 - 5.04T + 53T^{2} \)
59 \( 1 + 5.66T + 59T^{2} \)
61 \( 1 - 6.42T + 61T^{2} \)
67 \( 1 - 1.69T + 67T^{2} \)
71 \( 1 - 2.02T + 71T^{2} \)
73 \( 1 + 9.64T + 73T^{2} \)
79 \( 1 + 5.55T + 79T^{2} \)
83 \( 1 - 6.25T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.28337190029002535530303763496, −9.407212578747756788838488572971, −8.805268938365506362385183707347, −8.310600149549945089144559326403, −7.41577794896918790490720343717, −6.69960857252530947665157764552, −4.71726727247958300999847133889, −3.76367899357503732763504718450, −2.81008976254879807105378062159, −1.06488908327438773038967481836, 1.06488908327438773038967481836, 2.81008976254879807105378062159, 3.76367899357503732763504718450, 4.71726727247958300999847133889, 6.69960857252530947665157764552, 7.41577794896918790490720343717, 8.310600149549945089144559326403, 8.805268938365506362385183707347, 9.407212578747756788838488572971, 10.28337190029002535530303763496

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