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  + 2.17e15·2-s + 6.93e23·3-s + 2.21e30·4-s − 2.56e35·5-s + 1.51e39·6-s + 8.01e42·7-s − 7.08e44·8-s − 1.06e48·9-s − 5.58e50·10-s − 9.55e51·11-s + 1.53e54·12-s + 4.33e54·13-s + 1.74e58·14-s − 1.77e59·15-s − 7.14e60·16-s − 7.76e61·17-s − 2.31e63·18-s + 3.75e64·19-s − 5.66e65·20-s + 5.56e66·21-s − 2.08e67·22-s − 1.01e69·23-s − 4.91e68·24-s + 2.62e70·25-s + 9.45e69·26-s − 1.81e72·27-s + 1.77e73·28-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.557·3-s + 0.871·4-s − 1.29·5-s + 0.763·6-s + 1.68·7-s − 0.175·8-s − 0.688·9-s − 1.76·10-s − 0.245·11-s + 0.486·12-s + 0.0241·13-s + 2.30·14-s − 0.719·15-s − 1.11·16-s − 0.565·17-s − 0.942·18-s + 0.994·19-s − 1.12·20-s + 0.939·21-s − 0.335·22-s − 1.74·23-s − 0.0979·24-s + 0.665·25-s + 0.0330·26-s − 0.942·27-s + 1.46·28-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 \( 1 - 2.17e15T + 2.53e30T^{2} \)
3 \( 1 - 6.93e23T + 1.54e48T^{2} \)
5 \( 1 + 2.56e35T + 3.94e70T^{2} \)
7 \( 1 - 8.01e42T + 2.26e85T^{2} \)
11 \( 1 + 9.55e51T + 1.51e105T^{2} \)
13 \( 1 - 4.33e54T + 3.22e112T^{2} \)
17 \( 1 + 7.76e61T + 1.88e124T^{2} \)
19 \( 1 - 3.75e64T + 1.42e129T^{2} \)
23 \( 1 + 1.01e69T + 3.42e137T^{2} \)
29 \( 1 + 3.74e73T + 5.03e147T^{2} \)
31 \( 1 + 5.68e74T + 4.24e150T^{2} \)
37 \( 1 + 2.65e79T + 2.44e158T^{2} \)
41 \( 1 - 3.81e81T + 7.78e162T^{2} \)
43 \( 1 + 3.91e82T + 9.55e164T^{2} \)
47 \( 1 - 2.04e83T + 7.61e168T^{2} \)
53 \( 1 + 5.06e86T + 1.41e174T^{2} \)
59 \( 1 - 5.27e89T + 7.17e178T^{2} \)
61 \( 1 + 2.08e90T + 2.08e180T^{2} \)
67 \( 1 + 4.63e91T + 2.71e184T^{2} \)
71 \( 1 + 3.91e93T + 9.48e186T^{2} \)
73 \( 1 - 1.42e94T + 1.56e188T^{2} \)
79 \( 1 + 5.87e95T + 4.57e191T^{2} \)
83 \( 1 - 6.23e96T + 6.71e193T^{2} \)
89 \( 1 + 4.22e98T + 7.73e196T^{2} \)
97 \( 1 - 1.04e100T + 4.61e200T^{2} \)
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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

−13.83831344733983733217271901961, −11.97742793564205785717941294393, −11.29677745186060081272286594914, −8.558908603876670007740986979807, −7.58597400267031994504807971913, −5.49914897161061994479075920339, −4.38574778047642919774093686592, −3.43603241584958744486174212455, −2.03644198360498200132593859739, 0, 2.03644198360498200132593859739, 3.43603241584958744486174212455, 4.38574778047642919774093686592, 5.49914897161061994479075920339, 7.58597400267031994504807971913, 8.558908603876670007740986979807, 11.29677745186060081272286594914, 11.97742793564205785717941294393, 13.83831344733983733217271901961

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