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  + 2.56e15·2-s − 2.60e23·3-s + 4.03e30·4-s + 2.88e35·5-s − 6.68e38·6-s − 6.25e42·7-s + 3.85e45·8-s − 1.47e48·9-s + 7.40e50·10-s − 2.44e52·11-s − 1.05e54·12-s − 8.21e55·13-s − 1.60e58·14-s − 7.53e58·15-s − 3.62e59·16-s − 1.73e62·17-s − 3.78e63·18-s − 3.05e64·19-s + 1.16e66·20-s + 1.63e66·21-s − 6.26e67·22-s + 2.17e68·23-s − 1.00e69·24-s + 4.40e70·25-s − 2.10e71·26-s + 7.88e71·27-s − 2.52e73·28-s + ⋯
L(s)  = 1  + 1.61·2-s − 0.209·3-s + 1.59·4-s + 1.45·5-s − 0.337·6-s − 1.31·7-s + 0.954·8-s − 0.956·9-s + 2.34·10-s − 0.627·11-s − 0.333·12-s − 0.457·13-s − 2.11·14-s − 0.304·15-s − 0.0564·16-s − 1.26·17-s − 1.53·18-s − 0.808·19-s + 2.31·20-s + 0.275·21-s − 1.01·22-s + 0.370·23-s − 0.200·24-s + 1.11·25-s − 0.737·26-s + 0.410·27-s − 2.09·28-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 \( 1 - 2.56e15T + 2.53e30T^{2} \)
3 \( 1 + 2.60e23T + 1.54e48T^{2} \)
5 \( 1 - 2.88e35T + 3.94e70T^{2} \)
7 \( 1 + 6.25e42T + 2.26e85T^{2} \)
11 \( 1 + 2.44e52T + 1.51e105T^{2} \)
13 \( 1 + 8.21e55T + 3.22e112T^{2} \)
17 \( 1 + 1.73e62T + 1.88e124T^{2} \)
19 \( 1 + 3.05e64T + 1.42e129T^{2} \)
23 \( 1 - 2.17e68T + 3.42e137T^{2} \)
29 \( 1 - 1.39e74T + 5.03e147T^{2} \)
31 \( 1 + 3.23e75T + 4.24e150T^{2} \)
37 \( 1 - 1.43e79T + 2.44e158T^{2} \)
41 \( 1 - 4.24e81T + 7.78e162T^{2} \)
43 \( 1 - 1.21e82T + 9.55e164T^{2} \)
47 \( 1 + 3.72e84T + 7.61e168T^{2} \)
53 \( 1 + 1.31e87T + 1.41e174T^{2} \)
59 \( 1 + 2.92e89T + 7.17e178T^{2} \)
61 \( 1 + 4.65e89T + 2.08e180T^{2} \)
67 \( 1 - 1.13e92T + 2.71e184T^{2} \)
71 \( 1 + 1.95e93T + 9.48e186T^{2} \)
73 \( 1 + 9.13e93T + 1.56e188T^{2} \)
79 \( 1 + 4.27e95T + 4.57e191T^{2} \)
83 \( 1 - 1.25e97T + 6.71e193T^{2} \)
89 \( 1 - 1.65e98T + 7.73e196T^{2} \)
97 \( 1 - 1.54e100T + 4.61e200T^{2} \)
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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

−13.36552248261799854261674627339, −12.65368079237755217183639560011, −10.82077963014095848191775149913, −9.263024360124137157926574148563, −6.51887893445686011145051798849, −5.94508122584935265883163917558, −4.74586743505974520478766465134, −2.97352217396137672816372177901, −2.29952234914622231048095962944, 0, 2.29952234914622231048095962944, 2.97352217396137672816372177901, 4.74586743505974520478766465134, 5.94508122584935265883163917558, 6.51887893445686011145051798849, 9.263024360124137157926574148563, 10.82077963014095848191775149913, 12.65368079237755217183639560011, 13.36552248261799854261674627339

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