Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $101$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.38e15·2-s + 1.43e24·3-s + 3.15e30·4-s + 7.99e34·5-s − 3.42e39·6-s + 3.14e42·7-s − 1.46e45·8-s + 5.19e47·9-s − 1.90e50·10-s + 6.08e52·11-s + 4.52e54·12-s − 2.09e56·13-s − 7.50e57·14-s + 1.14e59·15-s − 4.48e60·16-s − 1.97e62·17-s − 1.23e63·18-s − 2.54e64·19-s + 2.51e65·20-s + 4.52e66·21-s − 1.45e68·22-s − 6.11e68·23-s − 2.10e69·24-s − 3.30e70·25-s + 5.00e71·26-s − 1.47e72·27-s + 9.91e72·28-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.15·3-s + 1.24·4-s + 0.402·5-s − 1.73·6-s + 0.661·7-s − 0.363·8-s + 0.335·9-s − 0.602·10-s + 1.56·11-s + 1.43·12-s − 1.16·13-s − 0.990·14-s + 0.465·15-s − 0.698·16-s − 1.43·17-s − 0.502·18-s − 0.672·19-s + 0.500·20-s + 0.764·21-s − 2.33·22-s − 1.04·23-s − 0.419·24-s − 0.838·25-s + 1.75·26-s − 0.767·27-s + 0.822·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(102-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+50.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(101\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :101/2),\ -1)\)

Particular Values

\(L(51)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{103}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.38e15T + 2.53e30T^{2} \)
3 \( 1 - 1.43e24T + 1.54e48T^{2} \)
5 \( 1 - 7.99e34T + 3.94e70T^{2} \)
7 \( 1 - 3.14e42T + 2.26e85T^{2} \)
11 \( 1 - 6.08e52T + 1.51e105T^{2} \)
13 \( 1 + 2.09e56T + 3.22e112T^{2} \)
17 \( 1 + 1.97e62T + 1.88e124T^{2} \)
19 \( 1 + 2.54e64T + 1.42e129T^{2} \)
23 \( 1 + 6.11e68T + 3.42e137T^{2} \)
29 \( 1 - 7.52e73T + 5.03e147T^{2} \)
31 \( 1 + 1.26e75T + 4.24e150T^{2} \)
37 \( 1 - 1.84e79T + 2.44e158T^{2} \)
41 \( 1 + 3.79e81T + 7.78e162T^{2} \)
43 \( 1 + 1.82e82T + 9.55e164T^{2} \)
47 \( 1 + 1.21e84T + 7.61e168T^{2} \)
53 \( 1 - 1.92e87T + 1.41e174T^{2} \)
59 \( 1 + 9.03e88T + 7.17e178T^{2} \)
61 \( 1 + 9.82e89T + 2.08e180T^{2} \)
67 \( 1 - 1.68e92T + 2.71e184T^{2} \)
71 \( 1 - 2.66e92T + 9.48e186T^{2} \)
73 \( 1 + 1.19e94T + 1.56e188T^{2} \)
79 \( 1 - 8.68e95T + 4.57e191T^{2} \)
83 \( 1 + 2.05e96T + 6.71e193T^{2} \)
89 \( 1 + 4.58e98T + 7.73e196T^{2} \)
97 \( 1 + 9.41e99T + 4.61e200T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.83543525086362276282104866803, −11.56423687984733212415701100264, −9.883925976031398564444302751297, −8.959148028139779382361218569643, −8.077839515092762504401350585889, −6.69424478243726127962893639230, −4.26343841374525617210445123056, −2.31181063522268616109911424924, −1.64705357458572343387092548283, 0, 1.64705357458572343387092548283, 2.31181063522268616109911424924, 4.26343841374525617210445123056, 6.69424478243726127962893639230, 8.077839515092762504401350585889, 8.959148028139779382361218569643, 9.883925976031398564444302751297, 11.56423687984733212415701100264, 13.83543525086362276282104866803

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