Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 61
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.80e8·2-s + 2.49e14·3-s − 1.34e18·4-s + 1.59e20·5-s + 2.44e23·6-s + 1.63e25·7-s − 3.57e27·8-s − 6.51e28·9-s + 1.56e29·10-s + 1.92e30·11-s − 3.34e32·12-s − 1.30e34·13-s + 1.60e34·14-s + 3.97e34·15-s − 4.08e35·16-s + 2.98e37·17-s − 6.38e37·18-s − 7.88e38·19-s − 2.14e38·20-s + 4.07e39·21-s + 1.88e39·22-s − 4.50e41·23-s − 8.91e41·24-s − 4.31e42·25-s − 1.28e43·26-s − 4.78e43·27-s − 2.19e43·28-s + ⋯
L(s)  = 1  + 0.645·2-s + 0.698·3-s − 0.583·4-s + 0.0765·5-s + 0.450·6-s + 0.274·7-s − 1.02·8-s − 0.512·9-s + 0.0494·10-s + 0.0332·11-s − 0.407·12-s − 1.38·13-s + 0.177·14-s + 0.0534·15-s − 0.0769·16-s + 0.884·17-s − 0.330·18-s − 0.784·19-s − 0.0446·20-s + 0.191·21-s + 0.0214·22-s − 1.31·23-s − 0.713·24-s − 0.994·25-s − 0.894·26-s − 1.05·27-s − 0.159·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(62-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(61\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :61/2),\ -1)\)
\(L(31)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{63}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 - 9.80e8T + 2.30e18T^{2} \)
3 \( 1 - 2.49e14T + 1.27e29T^{2} \)
5 \( 1 - 1.59e20T + 4.33e42T^{2} \)
7 \( 1 - 1.63e25T + 3.55e51T^{2} \)
11 \( 1 - 1.92e30T + 3.34e63T^{2} \)
13 \( 1 + 1.30e34T + 8.92e67T^{2} \)
17 \( 1 - 2.98e37T + 1.14e75T^{2} \)
19 \( 1 + 7.88e38T + 1.00e78T^{2} \)
23 \( 1 + 4.50e41T + 1.16e83T^{2} \)
29 \( 1 + 5.51e44T + 1.60e89T^{2} \)
31 \( 1 - 4.17e45T + 9.39e90T^{2} \)
37 \( 1 + 6.95e47T + 4.57e95T^{2} \)
41 \( 1 - 1.25e49T + 2.39e98T^{2} \)
43 \( 1 - 8.92e49T + 4.38e99T^{2} \)
47 \( 1 - 1.21e50T + 9.95e101T^{2} \)
53 \( 1 - 5.55e52T + 1.51e105T^{2} \)
59 \( 1 - 4.29e53T + 1.05e108T^{2} \)
61 \( 1 - 1.17e53T + 8.03e108T^{2} \)
67 \( 1 + 3.47e55T + 2.45e111T^{2} \)
71 \( 1 - 3.07e56T + 8.44e112T^{2} \)
73 \( 1 - 7.42e56T + 4.59e113T^{2} \)
79 \( 1 + 4.19e57T + 5.69e115T^{2} \)
83 \( 1 - 3.40e58T + 1.15e117T^{2} \)
89 \( 1 + 3.77e59T + 8.18e118T^{2} \)
97 \( 1 + 2.29e60T + 1.55e121T^{2} \)
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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

−17.40073597748457176333972756655, −14.85459368575987575054716742925, −13.89111881148042335247057293672, −12.17130866628167511468589375964, −9.613615850869011209910268229198, −8.035937676701038507713518680136, −5.58693134966324093625404413917, −3.97058657026526966063947217473, −2.41181351419019483043370514240, 0, 2.41181351419019483043370514240, 3.97058657026526966063947217473, 5.58693134966324093625404413917, 8.035937676701038507713518680136, 9.613615850869011209910268229198, 12.17130866628167511468589375964, 13.89111881148042335247057293672, 14.85459368575987575054716742925, 17.40073597748457176333972756655

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