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  + 2.03e15·2-s + 2.78e24·3-s − 6.00e30·4-s + 1.50e36·5-s + 5.67e39·6-s + 3.17e43·7-s − 3.28e46·8-s − 6.14e48·9-s + 3.07e51·10-s + 4.12e53·11-s − 1.67e55·12-s + 3.42e57·13-s + 6.46e58·14-s + 4.20e60·15-s − 5.95e60·16-s − 4.21e63·17-s − 1.24e64·18-s + 6.28e65·19-s − 9.05e66·20-s + 8.85e67·21-s + 8.39e68·22-s + 1.29e70·23-s − 9.15e70·24-s + 1.29e72·25-s + 6.96e72·26-s − 5.59e73·27-s − 1.90e74·28-s + ⋯
L(s)  = 1  + 0.638·2-s + 0.747·3-s − 0.591·4-s + 1.51·5-s + 0.477·6-s + 0.953·7-s − 1.01·8-s − 0.441·9-s + 0.970·10-s + 0.963·11-s − 0.442·12-s + 1.46·13-s + 0.609·14-s + 1.13·15-s − 0.0578·16-s − 1.80·17-s − 0.282·18-s + 0.876·19-s − 0.899·20-s + 0.712·21-s + 0.615·22-s + 0.965·23-s − 0.760·24-s + 1.30·25-s + 0.937·26-s − 1.07·27-s − 0.564·28-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  =  $(2,\ 1,\ (\ :103/2),\ 1)$
$L(52)$  $\approx$  $5.106166266$
$L(\frac12)$  $\approx$  $5.106166266$
$L(\frac{105}{2})$   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 \( 1 - 2.03e15T + 1.01e31T^{2} \)
3 \( 1 - 2.78e24T + 1.39e49T^{2} \)
5 \( 1 - 1.50e36T + 9.86e71T^{2} \)
7 \( 1 - 3.17e43T + 1.10e87T^{2} \)
11 \( 1 - 4.12e53T + 1.83e107T^{2} \)
13 \( 1 - 3.42e57T + 5.44e114T^{2} \)
17 \( 1 + 4.21e63T + 5.44e126T^{2} \)
19 \( 1 - 6.28e65T + 5.14e131T^{2} \)
23 \( 1 - 1.29e70T + 1.81e140T^{2} \)
29 \( 1 + 2.77e74T + 4.23e150T^{2} \)
31 \( 1 + 2.34e76T + 4.07e153T^{2} \)
37 \( 1 - 3.44e80T + 3.34e161T^{2} \)
41 \( 1 + 6.63e82T + 1.30e166T^{2} \)
43 \( 1 - 1.02e83T + 1.76e168T^{2} \)
47 \( 1 - 1.11e86T + 1.68e172T^{2} \)
53 \( 1 - 7.07e88T + 3.98e177T^{2} \)
59 \( 1 - 2.61e90T + 2.49e182T^{2} \)
61 \( 1 - 7.62e91T + 7.74e183T^{2} \)
67 \( 1 + 1.52e94T + 1.21e188T^{2} \)
71 \( 1 - 2.72e95T + 4.78e190T^{2} \)
73 \( 1 - 1.73e96T + 8.36e191T^{2} \)
79 \( 1 + 1.50e97T + 2.85e195T^{2} \)
83 \( 1 - 6.73e97T + 4.62e197T^{2} \)
89 \( 1 - 2.96e100T + 6.12e200T^{2} \)
97 \( 1 - 1.14e102T + 4.33e204T^{2} \)
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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

−13.85862850050302852461303290209, −13.37491091393082255231876107439, −11.24420329089941352446607042729, −9.234871532132719976604580286423, −8.674074854132539320723732447104, −6.30951755994007901788782959867, −5.18644165811606033499736683508, −3.77480185669757385586491601655, −2.35386395799161340447185020963, −1.16631250631130487402359409914, 1.16631250631130487402359409914, 2.35386395799161340447185020963, 3.77480185669757385586491601655, 5.18644165811606033499736683508, 6.30951755994007901788782959867, 8.674074854132539320723732447104, 9.234871532132719976604580286423, 11.24420329089941352446607042729, 13.37491091393082255231876107439, 13.85862850050302852461303290209

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