Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 89
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.81e13·2-s − 1.58e21·3-s + 1.69e27·4-s + 1.07e31·5-s + 7.64e34·6-s + 5.70e37·7-s − 5.17e40·8-s − 3.81e41·9-s − 5.18e44·10-s + 1.25e46·11-s − 2.69e48·12-s − 3.04e49·13-s − 2.74e51·14-s − 1.71e52·15-s + 1.44e54·16-s − 1.44e54·17-s + 1.83e55·18-s − 2.63e56·19-s + 1.82e58·20-s − 9.06e58·21-s − 6.05e59·22-s + 1.82e60·23-s + 8.23e61·24-s − 4.53e61·25-s + 1.46e63·26-s + 5.23e63·27-s + 9.66e64·28-s + ⋯
L(s)  = 1  − 1.93·2-s − 0.932·3-s + 2.73·4-s + 0.848·5-s + 1.80·6-s + 1.41·7-s − 3.36·8-s − 0.131·9-s − 1.63·10-s + 0.572·11-s − 2.55·12-s − 0.819·13-s − 2.72·14-s − 0.790·15-s + 3.76·16-s − 0.253·17-s + 0.253·18-s − 0.328·19-s + 2.32·20-s − 1.31·21-s − 1.10·22-s + 0.462·23-s + 3.13·24-s − 0.280·25-s + 1.58·26-s + 1.05·27-s + 3.86·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(90-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+89/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(89\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :89/2),\ -1)\)
\(L(45)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{91}{2})\)   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 \( 1 + 4.81e13T + 6.18e26T^{2} \)
3 \( 1 + 1.58e21T + 2.90e42T^{2} \)
5 \( 1 - 1.07e31T + 1.61e62T^{2} \)
7 \( 1 - 5.70e37T + 1.63e75T^{2} \)
11 \( 1 - 1.25e46T + 4.83e92T^{2} \)
13 \( 1 + 3.04e49T + 1.38e99T^{2} \)
17 \( 1 + 1.44e54T + 3.23e109T^{2} \)
19 \( 1 + 2.63e56T + 6.44e113T^{2} \)
23 \( 1 - 1.82e60T + 1.56e121T^{2} \)
29 \( 1 + 1.87e65T + 1.42e130T^{2} \)
31 \( 1 - 9.55e65T + 5.38e132T^{2} \)
37 \( 1 - 6.40e69T + 3.71e139T^{2} \)
41 \( 1 + 1.04e72T + 3.44e143T^{2} \)
43 \( 1 + 3.49e72T + 2.39e145T^{2} \)
47 \( 1 - 6.29e72T + 6.55e148T^{2} \)
53 \( 1 - 6.67e75T + 2.88e153T^{2} \)
59 \( 1 + 1.69e78T + 4.03e157T^{2} \)
61 \( 1 - 1.46e79T + 7.84e158T^{2} \)
67 \( 1 - 2.58e81T + 3.31e162T^{2} \)
71 \( 1 - 2.08e82T + 5.78e164T^{2} \)
73 \( 1 - 2.80e82T + 6.85e165T^{2} \)
79 \( 1 - 1.45e84T + 7.74e168T^{2} \)
83 \( 1 + 3.27e85T + 6.27e170T^{2} \)
89 \( 1 + 3.07e86T + 3.13e173T^{2} \)
97 \( 1 - 1.70e88T + 6.64e176T^{2} \)
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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

−14.84857644081875279050601119220, −11.76975237854152987114441787029, −10.93982418972054598244274876526, −9.600846531089496966527052970740, −8.231197368763280992121787890134, −6.74762281970422833159493819678, −5.40996499257468715029139143785, −2.21905934986599851932169852405, −1.29402578060258513574725849796, 0, 1.29402578060258513574725849796, 2.21905934986599851932169852405, 5.40996499257468715029139143785, 6.74762281970422833159493819678, 8.231197368763280992121787890134, 9.600846531089496966527052970740, 10.93982418972054598244274876526, 11.76975237854152987114441787029, 14.84857644081875279050601119220

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