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  + 2.88e7·2-s + 1.23e13·3-s − 3.51e16·4-s − 1.26e19·5-s + 3.57e20·6-s + 2.79e23·7-s − 2.05e24·8-s − 2.08e25·9-s − 3.64e26·10-s + 4.43e28·11-s − 4.36e29·12-s + 1.55e30·13-s + 8.06e30·14-s − 1.56e32·15-s + 1.20e33·16-s + 8.73e32·17-s − 6.00e32·18-s + 1.64e35·19-s + 4.45e35·20-s + 3.46e36·21-s + 1.27e36·22-s + 5.17e37·23-s − 2.54e37·24-s − 1.17e38·25-s + 4.46e37·26-s − 2.42e39·27-s − 9.85e39·28-s + ⋯
L(s)  = 1  + 0.151·2-s + 0.938·3-s − 0.976·4-s − 0.759·5-s + 0.142·6-s + 1.60·7-s − 0.300·8-s − 0.119·9-s − 0.115·10-s + 1.01·11-s − 0.916·12-s + 0.360·13-s + 0.244·14-s − 0.712·15-s + 0.931·16-s + 0.126·17-s − 0.0181·18-s + 1.12·19-s + 0.741·20-s + 1.51·21-s + 0.154·22-s + 1.84·23-s − 0.281·24-s − 0.423·25-s + 0.0547·26-s − 1.05·27-s − 1.57·28-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  =  $(2,\ 1,\ (\ :55/2),\ 1)$
$L(28)$  $\approx$  $2.43416$
$L(\frac12)$  $\approx$  $2.43416$
$L(\frac{57}{2})$   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 \( 1 - 2.88e7T + 3.60e16T^{2} \)
3 \( 1 - 1.23e13T + 1.74e26T^{2} \)
5 \( 1 + 1.26e19T + 2.77e38T^{2} \)
7 \( 1 - 2.79e23T + 3.02e46T^{2} \)
11 \( 1 - 4.43e28T + 1.89e57T^{2} \)
13 \( 1 - 1.55e30T + 1.84e61T^{2} \)
17 \( 1 - 8.73e32T + 4.72e67T^{2} \)
19 \( 1 - 1.64e35T + 2.14e70T^{2} \)
23 \( 1 - 5.17e37T + 7.85e74T^{2} \)
29 \( 1 - 1.58e40T + 2.70e80T^{2} \)
31 \( 1 - 5.89e40T + 1.05e82T^{2} \)
37 \( 1 + 1.50e43T + 1.78e86T^{2} \)
41 \( 1 - 2.49e44T + 5.04e88T^{2} \)
43 \( 1 + 2.64e44T + 6.93e89T^{2} \)
47 \( 1 + 1.75e46T + 9.23e91T^{2} \)
53 \( 1 + 7.37e46T + 6.84e94T^{2} \)
59 \( 1 - 2.54e48T + 2.49e97T^{2} \)
61 \( 1 + 1.07e49T + 1.56e98T^{2} \)
67 \( 1 - 1.06e50T + 2.71e100T^{2} \)
71 \( 1 + 1.17e51T + 6.59e101T^{2} \)
73 \( 1 - 2.92e50T + 3.03e102T^{2} \)
79 \( 1 - 2.26e52T + 2.34e104T^{2} \)
83 \( 1 - 3.10e52T + 3.54e105T^{2} \)
89 \( 1 - 4.89e53T + 1.64e107T^{2} \)
97 \( 1 - 4.57e53T + 1.87e109T^{2} \)
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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

−19.35943162974186792353821919685, −17.59332407787385497159822231607, −14.86048417439935935483518138056, −13.91279710831300133068693251366, −11.61267383697861039540824713270, −8.964199280923750619958448691525, −7.929393418680684726188677607875, −4.84686041653671905108901001592, −3.44084370216387245829347446565, −1.15076563335679149777486733227, 1.15076563335679149777486733227, 3.44084370216387245829347446565, 4.84686041653671905108901001592, 7.929393418680684726188677607875, 8.964199280923750619958448691525, 11.61267383697861039540824713270, 13.91279710831300133068693251366, 14.86048417439935935483518138056, 17.59332407787385497159822231607, 19.35943162974186792353821919685

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