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  − 1.58e15·2-s − 4.68e24·3-s − 7.63e30·4-s + 1.63e35·5-s + 7.41e39·6-s − 1.85e42·7-s + 2.81e46·8-s + 8.01e48·9-s − 2.58e50·10-s + 3.07e52·11-s + 3.57e55·12-s − 3.16e56·13-s + 2.94e57·14-s − 7.65e59·15-s + 3.28e61·16-s − 2.91e63·17-s − 1.26e64·18-s − 3.76e65·19-s − 1.24e66·20-s + 8.70e66·21-s − 4.86e67·22-s − 2.35e70·23-s − 1.31e71·24-s − 9.59e71·25-s + 5.00e71·26-s + 2.76e73·27-s + 1.41e73·28-s + ⋯
L(s)  = 1  − 0.497·2-s − 1.25·3-s − 0.752·4-s + 0.164·5-s + 0.624·6-s − 0.0558·7-s + 0.871·8-s + 0.576·9-s − 0.0817·10-s + 0.0717·11-s + 0.945·12-s − 0.135·13-s + 0.0277·14-s − 0.206·15-s + 0.319·16-s − 1.24·17-s − 0.286·18-s − 0.524·19-s − 0.123·20-s + 0.0700·21-s − 0.0356·22-s − 1.74·23-s − 1.09·24-s − 0.972·25-s + 0.0673·26-s + 0.532·27-s + 0.0420·28-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  =  $(2,\ 1,\ (\ :103/2),\ 1)$
$L(52)$  $\approx$  $0.1582654222$
$L(\frac12)$  $\approx$  $0.1582654222$
$L(\frac{105}{2})$   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 \( 1 + 1.58e15T + 1.01e31T^{2} \)
3 \( 1 + 4.68e24T + 1.39e49T^{2} \)
5 \( 1 - 1.63e35T + 9.86e71T^{2} \)
7 \( 1 + 1.85e42T + 1.10e87T^{2} \)
11 \( 1 - 3.07e52T + 1.83e107T^{2} \)
13 \( 1 + 3.16e56T + 5.44e114T^{2} \)
17 \( 1 + 2.91e63T + 5.44e126T^{2} \)
19 \( 1 + 3.76e65T + 5.14e131T^{2} \)
23 \( 1 + 2.35e70T + 1.81e140T^{2} \)
29 \( 1 - 2.04e75T + 4.23e150T^{2} \)
31 \( 1 + 6.25e76T + 4.07e153T^{2} \)
37 \( 1 + 1.13e81T + 3.34e161T^{2} \)
41 \( 1 - 7.57e82T + 1.30e166T^{2} \)
43 \( 1 + 8.64e82T + 1.76e168T^{2} \)
47 \( 1 + 1.08e86T + 1.68e172T^{2} \)
53 \( 1 + 4.88e88T + 3.98e177T^{2} \)
59 \( 1 + 8.65e89T + 2.49e182T^{2} \)
61 \( 1 - 6.27e91T + 7.74e183T^{2} \)
67 \( 1 - 1.07e94T + 1.21e188T^{2} \)
71 \( 1 + 6.83e93T + 4.78e190T^{2} \)
73 \( 1 - 1.38e96T + 8.36e191T^{2} \)
79 \( 1 - 6.37e97T + 2.85e195T^{2} \)
83 \( 1 + 1.02e99T + 4.62e197T^{2} \)
89 \( 1 + 2.25e100T + 6.12e200T^{2} \)
97 \( 1 + 2.30e102T + 4.33e204T^{2} \)
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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

−13.94284744933461953122930515628, −12.45254199412207295326658413087, −11.00018206251433318940289009782, −9.813460463677395691800668312301, −8.328101314359725369576812343446, −6.55563700632176043601502958714, −5.28157029139588675981565694259, −4.10382264182646781003876121231, −1.78970678691218207625078008999, −0.23945463581396790477328781855, 0.23945463581396790477328781855, 1.78970678691218207625078008999, 4.10382264182646781003876121231, 5.28157029139588675981565694259, 6.55563700632176043601502958714, 8.328101314359725369576812343446, 9.813460463677395691800668312301, 11.00018206251433318940289009782, 12.45254199412207295326658413087, 13.94284744933461953122930515628

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