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  − 1.34e10·2-s + 7.87e15·3-s + 3.28e19·4-s − 3.01e23·5-s − 1.05e26·6-s − 4.20e26·7-s + 1.54e30·8-s − 3.06e31·9-s + 4.04e33·10-s + 1.62e34·11-s + 2.59e35·12-s − 3.08e37·13-s + 5.65e36·14-s − 2.37e39·15-s − 2.55e40·16-s − 1.59e41·17-s + 4.11e41·18-s + 1.03e43·19-s − 9.91e42·20-s − 3.31e42·21-s − 2.17e44·22-s − 3.59e45·23-s + 1.21e46·24-s + 2.30e46·25-s + 4.14e47·26-s − 9.71e47·27-s − 1.38e46·28-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.818·3-s + 0.222·4-s − 1.15·5-s − 0.904·6-s − 0.0205·7-s + 0.859·8-s − 0.330·9-s + 1.28·10-s + 0.210·11-s + 0.182·12-s − 1.48·13-s + 0.0227·14-s − 0.947·15-s − 1.17·16-s − 0.960·17-s + 0.365·18-s + 1.50·19-s − 0.258·20-s − 0.0168·21-s − 0.232·22-s − 0.865·23-s + 0.703·24-s + 0.340·25-s + 1.64·26-s − 1.08·27-s − 0.00458·28-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  =  \((2,\ 1,\ (\ :67/2),\ 1)\)
\(L(34)\)  \(\approx\)  \(0.7305257436\)
\(L(\frac12)\)  \(\approx\)  \(0.7305257436\)
\(L(\frac{69}{2})\)   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 \( 1 + 1.34e10T + 1.47e20T^{2} \)
3 \( 1 - 7.87e15T + 9.27e31T^{2} \)
5 \( 1 + 3.01e23T + 6.77e46T^{2} \)
7 \( 1 + 4.20e26T + 4.18e56T^{2} \)
11 \( 1 - 1.62e34T + 5.93e69T^{2} \)
13 \( 1 + 3.08e37T + 4.30e74T^{2} \)
17 \( 1 + 1.59e41T + 2.75e82T^{2} \)
19 \( 1 - 1.03e43T + 4.74e85T^{2} \)
23 \( 1 + 3.59e45T + 1.72e91T^{2} \)
29 \( 1 - 1.54e49T + 9.56e97T^{2} \)
31 \( 1 - 1.52e50T + 8.34e99T^{2} \)
37 \( 1 - 1.16e52T + 1.17e105T^{2} \)
41 \( 1 - 1.50e54T + 1.13e108T^{2} \)
43 \( 1 - 1.43e54T + 2.76e109T^{2} \)
47 \( 1 - 9.41e55T + 1.07e112T^{2} \)
53 \( 1 - 4.49e57T + 3.36e115T^{2} \)
59 \( 1 - 2.81e59T + 4.43e118T^{2} \)
61 \( 1 + 4.42e59T + 4.14e119T^{2} \)
67 \( 1 - 9.71e60T + 2.22e122T^{2} \)
71 \( 1 + 3.66e61T + 1.08e124T^{2} \)
73 \( 1 + 1.52e61T + 6.96e124T^{2} \)
79 \( 1 + 2.59e63T + 1.38e127T^{2} \)
83 \( 1 + 2.25e64T + 3.78e128T^{2} \)
89 \( 1 + 2.92e64T + 4.06e130T^{2} \)
97 \( 1 + 8.79e65T + 1.29e133T^{2} \)
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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

−17.59882734512744097320727731167, −15.81448367747186753295575471035, −14.07728569605045425316424108796, −11.73931573546291553117159630696, −9.755535798977241030557905771745, −8.391833007584314591458868850339, −7.43183866614746778765322778546, −4.38977130502919495322892275986, −2.60914844321186924080308460842, −0.60391611597935734051522552277, 0.60391611597935734051522552277, 2.60914844321186924080308460842, 4.38977130502919495322892275986, 7.43183866614746778765322778546, 8.391833007584314591458868850339, 9.755535798977241030557905771745, 11.73931573546291553117159630696, 14.07728569605045425316424108796, 15.81448367747186753295575471035, 17.59882734512744097320727731167

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