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  − 6.55e8·2-s + 1.58e14·3-s − 1.46e17·4-s + 7.23e20·5-s − 1.04e23·6-s + 1.08e25·7-s + 4.74e26·8-s + 1.10e28·9-s − 4.74e29·10-s − 2.58e30·11-s − 2.32e31·12-s − 5.56e32·13-s − 7.12e33·14-s + 1.14e35·15-s − 2.26e35·16-s + 1.35e35·17-s − 7.26e36·18-s + 4.79e37·19-s − 1.06e38·20-s + 1.72e39·21-s + 1.69e39·22-s − 3.11e39·23-s + 7.52e40·24-s + 3.50e41·25-s + 3.65e41·26-s − 4.84e41·27-s − 1.59e42·28-s + ⋯
L(s)  = 1  − 0.863·2-s + 1.33·3-s − 0.254·4-s + 1.73·5-s − 1.15·6-s + 1.27·7-s + 1.08·8-s + 0.784·9-s − 1.50·10-s − 0.491·11-s − 0.339·12-s − 0.766·13-s − 1.10·14-s + 2.32·15-s − 0.681·16-s + 0.0681·17-s − 0.677·18-s + 0.907·19-s − 0.441·20-s + 1.70·21-s + 0.424·22-s − 0.210·23-s + 1.44·24-s + 2.01·25-s + 0.661·26-s − 0.288·27-s − 0.324·28-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  =  $(2,\ 1,\ (\ :59/2),\ 1)$
$L(30)$  $\approx$  $2.61260$
$L(\frac12)$  $\approx$  $2.61260$
$L(\frac{61}{2})$   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 \( 1 + 6.55e8T + 5.76e17T^{2} \)
3 \( 1 - 1.58e14T + 1.41e28T^{2} \)
5 \( 1 - 7.23e20T + 1.73e41T^{2} \)
7 \( 1 - 1.08e25T + 7.25e49T^{2} \)
11 \( 1 + 2.58e30T + 2.76e61T^{2} \)
13 \( 1 + 5.56e32T + 5.28e65T^{2} \)
17 \( 1 - 1.35e35T + 3.94e72T^{2} \)
19 \( 1 - 4.79e37T + 2.79e75T^{2} \)
23 \( 1 + 3.11e39T + 2.19e80T^{2} \)
29 \( 1 - 1.44e43T + 1.91e86T^{2} \)
31 \( 1 + 1.86e43T + 9.78e87T^{2} \)
37 \( 1 - 1.60e46T + 3.34e92T^{2} \)
41 \( 1 + 6.72e47T + 1.42e95T^{2} \)
43 \( 1 - 1.54e48T + 2.36e96T^{2} \)
47 \( 1 - 4.96e48T + 4.50e98T^{2} \)
53 \( 1 - 3.89e50T + 5.39e101T^{2} \)
59 \( 1 + 2.34e52T + 3.02e104T^{2} \)
61 \( 1 + 7.62e51T + 2.16e105T^{2} \)
67 \( 1 - 2.92e53T + 5.47e107T^{2} \)
71 \( 1 - 2.90e54T + 1.67e109T^{2} \)
73 \( 1 + 6.77e54T + 8.63e109T^{2} \)
79 \( 1 + 1.07e56T + 9.12e111T^{2} \)
83 \( 1 - 5.08e56T + 1.68e113T^{2} \)
89 \( 1 - 3.31e57T + 1.03e115T^{2} \)
97 \( 1 - 7.26e58T + 1.65e117T^{2} \)
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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

−18.37118645506379680051992049775, −17.33229663480126584959774542782, −14.40088006701548876183634812640, −13.58164995328430937867782174192, −10.15477590426761803507527989292, −9.060582200406625685943185371625, −7.80551940878668052863160991095, −5.03166302061618668622875081168, −2.43010475932392204575042668200, −1.39111825076257501596238050121, 1.39111825076257501596238050121, 2.43010475932392204575042668200, 5.03166302061618668622875081168, 7.80551940878668052863160991095, 9.060582200406625685943185371625, 10.15477590426761803507527989292, 13.58164995328430937867782174192, 14.40088006701548876183634812640, 17.33229663480126584959774542782, 18.37118645506379680051992049775

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