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  − 6.27e8·2-s − 6.22e14·3-s − 1.91e18·4-s + 2.33e20·5-s + 3.90e23·6-s + 6.25e25·7-s + 2.64e27·8-s + 2.59e29·9-s − 1.46e29·10-s − 9.05e31·11-s + 1.18e33·12-s + 7.79e33·13-s − 3.92e34·14-s − 1.45e35·15-s + 2.74e36·16-s − 1.64e37·17-s − 1.63e38·18-s + 1.45e39·19-s − 4.47e38·20-s − 3.89e40·21-s + 5.68e40·22-s − 1.28e41·23-s − 1.64e42·24-s − 4.28e42·25-s − 4.89e42·26-s − 8.26e43·27-s − 1.19e44·28-s + ⋯
L(s)  = 1  − 0.413·2-s − 1.74·3-s − 0.829·4-s + 0.112·5-s + 0.721·6-s + 1.04·7-s + 0.756·8-s + 2.04·9-s − 0.0464·10-s − 1.56·11-s + 1.44·12-s + 0.824·13-s − 0.433·14-s − 0.196·15-s + 0.516·16-s − 0.485·17-s − 0.845·18-s + 1.44·19-s − 0.0931·20-s − 1.83·21-s + 0.646·22-s − 0.377·23-s − 1.31·24-s − 0.987·25-s − 0.341·26-s − 1.82·27-s − 0.869·28-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 \( 1 + 6.27e8T + 2.30e18T^{2} \)
3 \( 1 + 6.22e14T + 1.27e29T^{2} \)
5 \( 1 - 2.33e20T + 4.33e42T^{2} \)
7 \( 1 - 6.25e25T + 3.55e51T^{2} \)
11 \( 1 + 9.05e31T + 3.34e63T^{2} \)
13 \( 1 - 7.79e33T + 8.92e67T^{2} \)
17 \( 1 + 1.64e37T + 1.14e75T^{2} \)
19 \( 1 - 1.45e39T + 1.00e78T^{2} \)
23 \( 1 + 1.28e41T + 1.16e83T^{2} \)
29 \( 1 - 1.23e44T + 1.60e89T^{2} \)
31 \( 1 - 2.17e45T + 9.39e90T^{2} \)
37 \( 1 - 5.13e47T + 4.57e95T^{2} \)
41 \( 1 - 1.44e49T + 2.39e98T^{2} \)
43 \( 1 + 4.68e49T + 4.38e99T^{2} \)
47 \( 1 + 1.55e51T + 9.95e101T^{2} \)
53 \( 1 - 2.61e52T + 1.51e105T^{2} \)
59 \( 1 - 5.85e53T + 1.05e108T^{2} \)
61 \( 1 + 6.99e53T + 8.03e108T^{2} \)
67 \( 1 + 5.65e55T + 2.45e111T^{2} \)
71 \( 1 + 4.76e56T + 8.44e112T^{2} \)
73 \( 1 - 4.87e56T + 4.59e113T^{2} \)
79 \( 1 + 6.21e57T + 5.69e115T^{2} \)
83 \( 1 + 1.18e58T + 1.15e117T^{2} \)
89 \( 1 + 8.44e58T + 8.18e118T^{2} \)
97 \( 1 - 8.57e59T + 1.55e121T^{2} \)
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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

−17.80721841780138574368475006020, −16.06287245538794194153472077784, −13.34171173481475114819743477979, −11.44898793875203414420182295781, −10.17967125104493943940457901527, −7.86762902956605463769700265963, −5.61929033798567742138441997144, −4.63481380677916661648876098095, −1.23948223022818828095323826042, 0, 1.23948223022818828095323826042, 4.63481380677916661648876098095, 5.61929033798567742138441997144, 7.86762902956605463769700265963, 10.17967125104493943940457901527, 11.44898793875203414420182295781, 13.34171173481475114819743477979, 16.06287245538794194153472077784, 17.80721841780138574368475006020

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