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  − 2.56e14·2-s − 3.45e23·3-s − 2.46e30·4-s + 1.16e35·5-s + 8.87e37·6-s + 2.79e42·7-s + 1.28e45·8-s − 1.42e48·9-s − 2.98e49·10-s − 9.95e51·11-s + 8.53e53·12-s − 1.44e56·13-s − 7.18e56·14-s − 4.01e58·15-s + 5.93e60·16-s + 1.59e62·17-s + 3.66e62·18-s + 1.62e64·19-s − 2.86e65·20-s − 9.66e65·21-s + 2.55e66·22-s + 7.65e68·23-s − 4.44e68·24-s − 2.59e70·25-s + 3.70e70·26-s + 1.02e72·27-s − 6.90e72·28-s + ⋯
L(s)  = 1  − 0.161·2-s − 0.277·3-s − 0.973·4-s + 0.584·5-s + 0.0448·6-s + 0.587·7-s + 0.318·8-s − 0.922·9-s − 0.0943·10-s − 0.255·11-s + 0.270·12-s − 0.804·13-s − 0.0948·14-s − 0.162·15-s + 0.922·16-s + 1.16·17-s + 0.148·18-s + 0.429·19-s − 0.569·20-s − 0.163·21-s + 0.0412·22-s + 1.30·23-s − 0.0884·24-s − 0.657·25-s + 0.129·26-s + 0.534·27-s − 0.572·28-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 \( 1 + 2.56e14T + 2.53e30T^{2} \)
3 \( 1 + 3.45e23T + 1.54e48T^{2} \)
5 \( 1 - 1.16e35T + 3.94e70T^{2} \)
7 \( 1 - 2.79e42T + 2.26e85T^{2} \)
11 \( 1 + 9.95e51T + 1.51e105T^{2} \)
13 \( 1 + 1.44e56T + 3.22e112T^{2} \)
17 \( 1 - 1.59e62T + 1.88e124T^{2} \)
19 \( 1 - 1.62e64T + 1.42e129T^{2} \)
23 \( 1 - 7.65e68T + 3.42e137T^{2} \)
29 \( 1 + 7.16e72T + 5.03e147T^{2} \)
31 \( 1 - 1.90e75T + 4.24e150T^{2} \)
37 \( 1 + 1.10e79T + 2.44e158T^{2} \)
41 \( 1 + 1.87e81T + 7.78e162T^{2} \)
43 \( 1 - 9.41e81T + 9.55e164T^{2} \)
47 \( 1 - 3.14e84T + 7.61e168T^{2} \)
53 \( 1 + 4.00e86T + 1.41e174T^{2} \)
59 \( 1 + 2.34e89T + 7.17e178T^{2} \)
61 \( 1 + 4.02e89T + 2.08e180T^{2} \)
67 \( 1 + 3.05e92T + 2.71e184T^{2} \)
71 \( 1 + 5.37e93T + 9.48e186T^{2} \)
73 \( 1 + 2.11e94T + 1.56e188T^{2} \)
79 \( 1 - 8.91e95T + 4.57e191T^{2} \)
83 \( 1 - 7.50e96T + 6.71e193T^{2} \)
89 \( 1 + 3.83e98T + 7.73e196T^{2} \)
97 \( 1 - 1.03e100T + 4.61e200T^{2} \)
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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

−13.63839064707696521067735240696, −12.02216140305686714374739542125, −10.33326108398342959937778506038, −9.070094047322065456030915196540, −7.73703771264655209332127773026, −5.67772388008828555170485991535, −4.82698619144678625789844239724, −3.00220144996402089317871187059, −1.28682287481686156955760249788, 0, 1.28682287481686156955760249788, 3.00220144996402089317871187059, 4.82698619144678625789844239724, 5.67772388008828555170485991535, 7.73703771264655209332127773026, 9.070094047322065456030915196540, 10.33326108398342959937778506038, 12.02216140305686714374739542125, 13.63839064707696521067735240696

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