Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 67
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.43e9·2-s + 1.56e16·3-s − 7.64e19·4-s + 3.83e23·5-s + 1.32e26·6-s − 1.75e27·7-s − 1.88e30·8-s + 1.52e32·9-s + 3.23e33·10-s + 1.38e35·11-s − 1.19e36·12-s − 6.45e36·13-s − 1.48e37·14-s + 6.00e39·15-s − 4.64e39·16-s − 8.51e40·17-s + 1.28e42·18-s − 5.19e42·19-s − 2.93e43·20-s − 2.75e43·21-s + 1.16e45·22-s + 2.49e45·23-s − 2.96e46·24-s + 7.92e46·25-s − 5.44e46·26-s + 9.41e47·27-s + 1.34e47·28-s + ⋯
L(s)  = 1  + 0.694·2-s + 1.62·3-s − 0.518·4-s + 1.47·5-s + 1.12·6-s − 0.0858·7-s − 1.05·8-s + 1.64·9-s + 1.02·10-s + 1.80·11-s − 0.843·12-s − 0.311·13-s − 0.0595·14-s + 2.39·15-s − 0.213·16-s − 0.512·17-s + 1.14·18-s − 0.754·19-s − 0.763·20-s − 0.139·21-s + 1.25·22-s + 0.602·23-s − 1.71·24-s + 1.16·25-s − 0.215·26-s + 1.05·27-s + 0.0444·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(68-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+67/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(67\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1,\ (\ :67/2),\ 1)\)
\(L(34)\)  \(\approx\)  \(5.454300640\)
\(L(\frac12)\)  \(\approx\)  \(5.454300640\)
\(L(\frac{69}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 - 8.43e9T + 1.47e20T^{2} \)
3 \( 1 - 1.56e16T + 9.27e31T^{2} \)
5 \( 1 - 3.83e23T + 6.77e46T^{2} \)
7 \( 1 + 1.75e27T + 4.18e56T^{2} \)
11 \( 1 - 1.38e35T + 5.93e69T^{2} \)
13 \( 1 + 6.45e36T + 4.30e74T^{2} \)
17 \( 1 + 8.51e40T + 2.75e82T^{2} \)
19 \( 1 + 5.19e42T + 4.74e85T^{2} \)
23 \( 1 - 2.49e45T + 1.72e91T^{2} \)
29 \( 1 - 1.20e49T + 9.56e97T^{2} \)
31 \( 1 + 6.23e49T + 8.34e99T^{2} \)
37 \( 1 - 3.56e51T + 1.17e105T^{2} \)
41 \( 1 - 4.51e53T + 1.13e108T^{2} \)
43 \( 1 - 6.41e54T + 2.76e109T^{2} \)
47 \( 1 + 1.03e56T + 1.07e112T^{2} \)
53 \( 1 + 9.61e57T + 3.36e115T^{2} \)
59 \( 1 + 4.77e58T + 4.43e118T^{2} \)
61 \( 1 + 2.98e58T + 4.14e119T^{2} \)
67 \( 1 + 2.03e61T + 2.22e122T^{2} \)
71 \( 1 + 6.57e61T + 1.08e124T^{2} \)
73 \( 1 + 1.20e62T + 6.96e124T^{2} \)
79 \( 1 - 2.11e63T + 1.38e127T^{2} \)
83 \( 1 + 3.25e64T + 3.78e128T^{2} \)
89 \( 1 - 6.19e64T + 4.06e130T^{2} \)
97 \( 1 + 9.58e65T + 1.29e133T^{2} \)
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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

−17.53451955114928655257002183496, −14.65779117134789917429773757776, −14.03740373250213738095153796765, −12.88744124641297417452334233666, −9.554787559742110402724174000518, −8.858675758657522720164921571811, −6.37818207318158337389829469495, −4.36670364333491102955122907493, −2.94488524404003807235366340415, −1.59737227155102043920021945378, 1.59737227155102043920021945378, 2.94488524404003807235366340415, 4.36670364333491102955122907493, 6.37818207318158337389829469495, 8.858675758657522720164921571811, 9.554787559742110402724174000518, 12.88744124641297417452334233666, 14.03740373250213738095153796765, 14.65779117134789917429773757776, 17.53451955114928655257002183496

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