Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 55
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.25e8·2-s + 5.98e12·3-s + 7.01e16·4-s + 2.73e19·5-s + 1.94e21·6-s − 1.10e23·7-s + 1.11e25·8-s − 1.38e26·9-s + 8.91e27·10-s + 4.91e28·11-s + 4.19e29·12-s − 5.50e30·13-s − 3.59e31·14-s + 1.63e32·15-s + 1.09e33·16-s + 6.74e33·17-s − 4.51e34·18-s + 4.63e34·19-s + 1.92e36·20-s − 6.59e35·21-s + 1.60e37·22-s − 2.33e37·23-s + 6.65e37·24-s + 4.71e38·25-s − 1.79e39·26-s − 1.87e39·27-s − 7.74e39·28-s + ⋯
L(s)  = 1  + 1.71·2-s + 0.452·3-s + 1.94·4-s + 1.64·5-s + 0.777·6-s − 0.634·7-s + 1.62·8-s − 0.794·9-s + 2.82·10-s + 1.13·11-s + 0.882·12-s − 1.28·13-s − 1.08·14-s + 0.743·15-s + 0.845·16-s + 0.980·17-s − 1.36·18-s + 0.316·19-s + 3.19·20-s − 0.287·21-s + 1.94·22-s − 0.833·23-s + 0.736·24-s + 1.69·25-s − 2.19·26-s − 0.812·27-s − 1.23·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(56-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+55/2) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(55\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :55/2),\ 1)$
$L(28)$  $\approx$  $6.66977$
$L(\frac12)$  $\approx$  $6.66977$
$L(\frac{57}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 2.
$p$$F_p$
good2 \( 1 - 3.25e8T + 3.60e16T^{2} \)
3 \( 1 - 5.98e12T + 1.74e26T^{2} \)
5 \( 1 - 2.73e19T + 2.77e38T^{2} \)
7 \( 1 + 1.10e23T + 3.02e46T^{2} \)
11 \( 1 - 4.91e28T + 1.89e57T^{2} \)
13 \( 1 + 5.50e30T + 1.84e61T^{2} \)
17 \( 1 - 6.74e33T + 4.72e67T^{2} \)
19 \( 1 - 4.63e34T + 2.14e70T^{2} \)
23 \( 1 + 2.33e37T + 7.85e74T^{2} \)
29 \( 1 - 8.29e39T + 2.70e80T^{2} \)
31 \( 1 + 2.22e40T + 1.05e82T^{2} \)
37 \( 1 + 8.62e42T + 1.78e86T^{2} \)
41 \( 1 + 1.28e44T + 5.04e88T^{2} \)
43 \( 1 + 7.80e44T + 6.93e89T^{2} \)
47 \( 1 - 2.24e45T + 9.23e91T^{2} \)
53 \( 1 - 6.54e46T + 6.84e94T^{2} \)
59 \( 1 + 4.81e48T + 2.49e97T^{2} \)
61 \( 1 + 1.00e48T + 1.56e98T^{2} \)
67 \( 1 - 2.13e50T + 2.71e100T^{2} \)
71 \( 1 - 9.56e50T + 6.59e101T^{2} \)
73 \( 1 - 1.10e51T + 3.03e102T^{2} \)
79 \( 1 - 2.07e52T + 2.34e104T^{2} \)
83 \( 1 + 1.06e53T + 3.54e105T^{2} \)
89 \( 1 - 7.48e53T + 1.64e107T^{2} \)
97 \( 1 + 5.59e53T + 1.87e109T^{2} \)
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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

−20.01209228210802239140331363890, −16.95845876810857549969086353216, −14.52645529032248737866988910771, −13.80465243210394443021771861078, −12.20664586550508961397681257035, −9.671107198526613643329808404749, −6.48228350335189907889910527584, −5.32510389581082330361180733567, −3.26013001793659364410915219565, −2.04453281596194447419305485636, 2.04453281596194447419305485636, 3.26013001793659364410915219565, 5.32510389581082330361180733567, 6.48228350335189907889910527584, 9.671107198526613643329808404749, 12.20664586550508961397681257035, 13.80465243210394443021771861078, 14.52645529032248737866988910771, 16.95845876810857549969086353216, 20.01209228210802239140331363890

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