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  + 0.258·2-s + 2.67·3-s − 1.93·4-s − 2.46·5-s + 0.692·6-s + 1.88·7-s − 1.01·8-s + 4.15·9-s − 0.637·10-s + 2.17·11-s − 5.17·12-s + 5.79·13-s + 0.486·14-s − 6.58·15-s + 3.60·16-s + 5.77·17-s + 1.07·18-s − 2.25·19-s + 4.75·20-s + 5.03·21-s + 0.562·22-s − 2.33·23-s − 2.72·24-s + 1.05·25-s + 1.49·26-s + 3.10·27-s − 3.63·28-s + ⋯
L(s)  = 1  + 0.183·2-s + 1.54·3-s − 0.966·4-s − 1.10·5-s + 0.282·6-s + 0.710·7-s − 0.359·8-s + 1.38·9-s − 0.201·10-s + 0.655·11-s − 1.49·12-s + 1.60·13-s + 0.130·14-s − 1.70·15-s + 0.900·16-s + 1.40·17-s + 0.253·18-s − 0.517·19-s + 1.06·20-s + 1.09·21-s + 0.120·22-s − 0.487·23-s − 0.556·24-s + 0.211·25-s + 0.293·26-s + 0.597·27-s − 0.687·28-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  =  $(2,\ 619,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.12948$
$L(\frac12)$  $\approx$  $2.12948$
$L(\frac{3}{2})$   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 \( 1 - 0.258T + 2T^{2} \)
3 \( 1 - 2.67T + 3T^{2} \)
5 \( 1 + 2.46T + 5T^{2} \)
7 \( 1 - 1.88T + 7T^{2} \)
11 \( 1 - 2.17T + 11T^{2} \)
13 \( 1 - 5.79T + 13T^{2} \)
17 \( 1 - 5.77T + 17T^{2} \)
19 \( 1 + 2.25T + 19T^{2} \)
23 \( 1 + 2.33T + 23T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 + 9.67T + 31T^{2} \)
37 \( 1 - 8.94T + 37T^{2} \)
41 \( 1 + 1.23T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 - 6.19T + 47T^{2} \)
53 \( 1 + 0.0669T + 53T^{2} \)
59 \( 1 + 8.20T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 0.238T + 67T^{2} \)
71 \( 1 - 2.25T + 71T^{2} \)
73 \( 1 + 9.40T + 73T^{2} \)
79 \( 1 - 1.71T + 79T^{2} \)
83 \( 1 + 1.43T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 - 9.35T + 97T^{2} \)
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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

−10.48859213906703268210506230728, −9.430518132756196774581114710816, −8.638753132551186059172745897804, −8.181480783618787084504496208097, −7.61317455521247603539238586742, −6.07511155503227349966165091712, −4.61180467888344143505041662948, −3.80480367954781856971379722944, −3.28568403240420321156010004962, −1.36830780614071780582187215135, 1.36830780614071780582187215135, 3.28568403240420321156010004962, 3.80480367954781856971379722944, 4.61180467888344143505041662948, 6.07511155503227349966165091712, 7.61317455521247603539238586742, 8.181480783618787084504496208097, 8.638753132551186059172745897804, 9.430518132756196774581114710816, 10.48859213906703268210506230728

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