Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 103
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.46e15·2-s − 1.30e24·3-s + 1.96e31·4-s − 5.30e35·5-s + 7.10e39·6-s + 2.61e43·7-s − 5.20e46·8-s − 1.22e49·9-s + 2.89e51·10-s + 4.50e53·11-s − 2.55e55·12-s + 2.00e57·13-s − 1.43e59·14-s + 6.90e59·15-s + 8.46e61·16-s + 3.55e63·17-s + 6.67e64·18-s + 1.06e66·19-s − 1.04e67·20-s − 3.40e67·21-s − 2.45e69·22-s + 1.51e70·23-s + 6.77e70·24-s − 7.04e71·25-s − 1.09e73·26-s + 3.40e73·27-s + 5.15e74·28-s + ⋯
L(s)  = 1  − 1.71·2-s − 0.348·3-s + 1.93·4-s − 0.534·5-s + 0.597·6-s + 0.786·7-s − 1.61·8-s − 0.878·9-s + 0.916·10-s + 1.05·11-s − 0.676·12-s + 0.857·13-s − 1.34·14-s + 0.186·15-s + 0.823·16-s + 1.52·17-s + 1.50·18-s + 1.47·19-s − 1.03·20-s − 0.274·21-s − 1.80·22-s + 1.12·23-s + 0.562·24-s − 0.714·25-s − 1.47·26-s + 0.655·27-s + 1.52·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(104-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+51.5) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(103\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :103/2),\ 1)$
$L(52)$  $\approx$  $0.8749951250$
$L(\frac12)$  $\approx$  $0.8749951250$
$L(\frac{105}{2})$   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 \( 1 + 5.46e15T + 1.01e31T^{2} \)
3 \( 1 + 1.30e24T + 1.39e49T^{2} \)
5 \( 1 + 5.30e35T + 9.86e71T^{2} \)
7 \( 1 - 2.61e43T + 1.10e87T^{2} \)
11 \( 1 - 4.50e53T + 1.83e107T^{2} \)
13 \( 1 - 2.00e57T + 5.44e114T^{2} \)
17 \( 1 - 3.55e63T + 5.44e126T^{2} \)
19 \( 1 - 1.06e66T + 5.14e131T^{2} \)
23 \( 1 - 1.51e70T + 1.81e140T^{2} \)
29 \( 1 + 1.95e75T + 4.23e150T^{2} \)
31 \( 1 + 6.33e76T + 4.07e153T^{2} \)
37 \( 1 + 3.73e80T + 3.34e161T^{2} \)
41 \( 1 - 1.79e83T + 1.30e166T^{2} \)
43 \( 1 + 4.86e83T + 1.76e168T^{2} \)
47 \( 1 + 1.62e86T + 1.68e172T^{2} \)
53 \( 1 - 7.08e88T + 3.98e177T^{2} \)
59 \( 1 + 2.20e90T + 2.49e182T^{2} \)
61 \( 1 - 4.53e91T + 7.74e183T^{2} \)
67 \( 1 - 1.42e94T + 1.21e188T^{2} \)
71 \( 1 - 1.63e95T + 4.78e190T^{2} \)
73 \( 1 + 1.05e96T + 8.36e191T^{2} \)
79 \( 1 + 7.93e97T + 2.85e195T^{2} \)
83 \( 1 - 5.44e98T + 4.62e197T^{2} \)
89 \( 1 + 3.14e100T + 6.12e200T^{2} \)
97 \( 1 - 6.67e101T + 4.33e204T^{2} \)
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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

−14.49588875008281467116106729797, −11.70265211969138772173430253792, −11.14357942641991722671572308843, −9.459995306227524420765984900665, −8.314445867155488188090328983035, −7.27363451192625312106361995270, −5.61373521161956082782118479309, −3.39208685451152238495680886104, −1.51236689024897907823367013330, −0.71731748811295139800188646784, 0.71731748811295139800188646784, 1.51236689024897907823367013330, 3.39208685451152238495680886104, 5.61373521161956082782118479309, 7.27363451192625312106361995270, 8.314445867155488188090328983035, 9.459995306227524420765984900665, 11.14357942641991722671572308843, 11.70265211969138772173430253792, 14.49588875008281467116106729797

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