Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 59
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.25e9·2-s + 2.00e14·3-s + 1.00e18·4-s − 4.05e20·5-s + 2.52e23·6-s + 2.66e24·7-s + 5.43e26·8-s + 2.62e28·9-s − 5.09e29·10-s + 7.79e30·11-s + 2.02e32·12-s + 1.35e32·13-s + 3.35e33·14-s − 8.14e34·15-s + 1.02e35·16-s − 2.91e36·17-s + 3.30e37·18-s − 6.13e36·19-s − 4.08e38·20-s + 5.35e38·21-s + 9.81e39·22-s − 2.47e39·23-s + 1.09e41·24-s − 9.34e39·25-s + 1.71e41·26-s + 2.43e42·27-s + 2.68e42·28-s + ⋯
L(s)  = 1  + 1.65·2-s + 1.69·3-s + 1.74·4-s − 0.972·5-s + 2.80·6-s + 0.312·7-s + 1.24·8-s + 1.85·9-s − 1.61·10-s + 1.48·11-s + 2.95·12-s + 0.187·13-s + 0.518·14-s − 1.64·15-s + 0.309·16-s − 1.46·17-s + 3.08·18-s − 0.115·19-s − 1.70·20-s + 0.528·21-s + 2.45·22-s − 0.167·23-s + 2.09·24-s − 0.0538·25-s + 0.310·26-s + 1.45·27-s + 0.546·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(60-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+59/2) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(59\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :59/2),\ 1)$
$L(30)$  $\approx$  $7.56809$
$L(\frac12)$  $\approx$  $7.56809$
$L(\frac{61}{2})$   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 \( 1 - 1.25e9T + 5.76e17T^{2} \)
3 \( 1 - 2.00e14T + 1.41e28T^{2} \)
5 \( 1 + 4.05e20T + 1.73e41T^{2} \)
7 \( 1 - 2.66e24T + 7.25e49T^{2} \)
11 \( 1 - 7.79e30T + 2.76e61T^{2} \)
13 \( 1 - 1.35e32T + 5.28e65T^{2} \)
17 \( 1 + 2.91e36T + 3.94e72T^{2} \)
19 \( 1 + 6.13e36T + 2.79e75T^{2} \)
23 \( 1 + 2.47e39T + 2.19e80T^{2} \)
29 \( 1 + 4.74e42T + 1.91e86T^{2} \)
31 \( 1 - 1.04e43T + 9.78e87T^{2} \)
37 \( 1 + 7.35e45T + 3.34e92T^{2} \)
41 \( 1 + 6.31e47T + 1.42e95T^{2} \)
43 \( 1 + 5.26e47T + 2.36e96T^{2} \)
47 \( 1 - 3.33e49T + 4.50e98T^{2} \)
53 \( 1 - 7.06e50T + 5.39e101T^{2} \)
59 \( 1 - 2.46e52T + 3.02e104T^{2} \)
61 \( 1 - 5.32e52T + 2.16e105T^{2} \)
67 \( 1 + 2.99e53T + 5.47e107T^{2} \)
71 \( 1 + 4.94e54T + 1.67e109T^{2} \)
73 \( 1 - 7.41e54T + 8.63e109T^{2} \)
79 \( 1 + 6.76e55T + 9.12e111T^{2} \)
83 \( 1 - 4.29e56T + 1.68e113T^{2} \)
89 \( 1 + 4.87e56T + 1.03e115T^{2} \)
97 \( 1 + 9.59e57T + 1.65e117T^{2} \)
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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

−19.54913415213792043428723289074, −15.53222988594647263138409618678, −14.60886506352968612888940197488, −13.42961960403389661708494348432, −11.71835413971820607145744416761, −8.698606407186667715640325704438, −6.93945325990773163531099102392, −4.26781622403418384419977402297, −3.55585079515958630395106621156, −2.02539191838660150640447250153, 2.02539191838660150640447250153, 3.55585079515958630395106621156, 4.26781622403418384419977402297, 6.93945325990773163531099102392, 8.698606407186667715640325704438, 11.71835413971820607145744416761, 13.42961960403389661708494348432, 14.60886506352968612888940197488, 15.53222988594647263138409618678, 19.54913415213792043428723289074

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