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  + 3.09e15·2-s − 9.30e23·3-s − 5.91e29·4-s − 7.61e35·5-s − 2.87e39·6-s − 3.39e43·7-s − 3.31e46·8-s − 1.30e49·9-s − 2.35e51·10-s − 1.54e53·11-s + 5.50e53·12-s − 6.78e56·13-s − 1.04e59·14-s + 7.08e59·15-s − 9.64e61·16-s + 3.47e63·17-s − 4.03e64·18-s − 3.44e64·19-s + 4.50e65·20-s + 3.15e67·21-s − 4.78e68·22-s − 5.84e69·23-s + 3.08e70·24-s − 4.06e71·25-s − 2.09e72·26-s + 2.50e73·27-s + 2.00e73·28-s + ⋯
L(s)  = 1  + 0.970·2-s − 0.249·3-s − 0.0583·4-s − 0.766·5-s − 0.242·6-s − 1.01·7-s − 1.02·8-s − 0.937·9-s − 0.743·10-s − 0.361·11-s + 0.0145·12-s − 0.290·13-s − 0.988·14-s + 0.191·15-s − 0.938·16-s + 1.49·17-s − 0.909·18-s − 0.0480·19-s + 0.0447·20-s + 0.254·21-s − 0.350·22-s − 0.434·23-s + 0.256·24-s − 0.412·25-s − 0.282·26-s + 0.483·27-s + 0.0594·28-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  =  $(2,\ 1,\ (\ :103/2),\ 1)$
$L(52)$  $\approx$  $0.8592002968$
$L(\frac12)$  $\approx$  $0.8592002968$
$L(\frac{105}{2})$   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 \( 1 - 3.09e15T + 1.01e31T^{2} \)
3 \( 1 + 9.30e23T + 1.39e49T^{2} \)
5 \( 1 + 7.61e35T + 9.86e71T^{2} \)
7 \( 1 + 3.39e43T + 1.10e87T^{2} \)
11 \( 1 + 1.54e53T + 1.83e107T^{2} \)
13 \( 1 + 6.78e56T + 5.44e114T^{2} \)
17 \( 1 - 3.47e63T + 5.44e126T^{2} \)
19 \( 1 + 3.44e64T + 5.14e131T^{2} \)
23 \( 1 + 5.84e69T + 1.81e140T^{2} \)
29 \( 1 - 3.50e73T + 4.23e150T^{2} \)
31 \( 1 - 1.38e76T + 4.07e153T^{2} \)
37 \( 1 - 1.03e81T + 3.34e161T^{2} \)
41 \( 1 + 1.61e83T + 1.30e166T^{2} \)
43 \( 1 + 2.10e84T + 1.76e168T^{2} \)
47 \( 1 - 1.62e86T + 1.68e172T^{2} \)
53 \( 1 + 6.47e88T + 3.98e177T^{2} \)
59 \( 1 + 7.57e90T + 2.49e182T^{2} \)
61 \( 1 + 4.94e91T + 7.74e183T^{2} \)
67 \( 1 - 1.38e94T + 1.21e188T^{2} \)
71 \( 1 - 3.57e95T + 4.78e190T^{2} \)
73 \( 1 - 8.65e95T + 8.36e191T^{2} \)
79 \( 1 + 9.98e96T + 2.85e195T^{2} \)
83 \( 1 + 8.18e98T + 4.62e197T^{2} \)
89 \( 1 + 7.53e99T + 6.12e200T^{2} \)
97 \( 1 - 3.33e102T + 4.33e204T^{2} \)
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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

−14.02356909803062198055849801875, −12.62381748292097611643865224667, −11.66906820814829994053014708486, −9.743739958062291093058982123269, −8.078946471701478388887178092055, −6.27148301612184631398673383049, −5.18555392853539861189669321254, −3.71757155189670630661587033188, −2.87955648715369835865194743466, −0.40526073681439154684249278264, 0.40526073681439154684249278264, 2.87955648715369835865194743466, 3.71757155189670630661587033188, 5.18555392853539861189669321254, 6.27148301612184631398673383049, 8.078946471701478388887178092055, 9.743739958062291093058982123269, 11.66906820814829994053014708486, 12.62381748292097611643865224667, 14.02356909803062198055849801875

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