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  − 1.03e13·2-s − 5.04e20·3-s − 5.11e26·4-s + 2.06e31·5-s + 5.24e33·6-s − 1.95e37·7-s + 1.17e40·8-s − 2.65e42·9-s − 2.14e44·10-s − 3.33e46·11-s + 2.58e47·12-s + 6.87e49·13-s + 2.02e50·14-s − 1.04e52·15-s + 1.94e53·16-s − 2.38e54·17-s + 2.75e55·18-s + 1.04e57·19-s − 1.05e58·20-s + 9.85e57·21-s + 3.46e59·22-s − 1.76e60·23-s − 5.92e60·24-s + 2.66e62·25-s − 7.14e62·26-s + 2.80e63·27-s + 9.97e63·28-s + ⋯
L(s)  = 1  − 0.417·2-s − 0.295·3-s − 0.825·4-s + 1.62·5-s + 0.123·6-s − 0.482·7-s + 0.762·8-s − 0.912·9-s − 0.679·10-s − 1.51·11-s + 0.244·12-s + 1.84·13-s + 0.201·14-s − 0.481·15-s + 0.507·16-s − 0.419·17-s + 0.380·18-s + 1.29·19-s − 1.34·20-s + 0.142·21-s + 0.634·22-s − 0.447·23-s − 0.225·24-s + 1.64·25-s − 0.772·26-s + 0.566·27-s + 0.398·28-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 \( 1 + 1.03e13T + 6.18e26T^{2} \)
3 \( 1 + 5.04e20T + 2.90e42T^{2} \)
5 \( 1 - 2.06e31T + 1.61e62T^{2} \)
7 \( 1 + 1.95e37T + 1.63e75T^{2} \)
11 \( 1 + 3.33e46T + 4.83e92T^{2} \)
13 \( 1 - 6.87e49T + 1.38e99T^{2} \)
17 \( 1 + 2.38e54T + 3.23e109T^{2} \)
19 \( 1 - 1.04e57T + 6.44e113T^{2} \)
23 \( 1 + 1.76e60T + 1.56e121T^{2} \)
29 \( 1 - 3.58e64T + 1.42e130T^{2} \)
31 \( 1 + 5.07e65T + 5.38e132T^{2} \)
37 \( 1 - 3.65e69T + 3.71e139T^{2} \)
41 \( 1 + 8.30e71T + 3.44e143T^{2} \)
43 \( 1 - 1.00e72T + 2.39e145T^{2} \)
47 \( 1 + 1.68e74T + 6.55e148T^{2} \)
53 \( 1 + 3.74e76T + 2.88e153T^{2} \)
59 \( 1 + 6.54e78T + 4.03e157T^{2} \)
61 \( 1 - 6.58e78T + 7.84e158T^{2} \)
67 \( 1 + 6.71e80T + 3.31e162T^{2} \)
71 \( 1 + 3.76e82T + 5.78e164T^{2} \)
73 \( 1 - 8.64e81T + 6.85e165T^{2} \)
79 \( 1 - 1.41e84T + 7.74e168T^{2} \)
83 \( 1 + 6.96e84T + 6.27e170T^{2} \)
89 \( 1 + 1.63e86T + 3.13e173T^{2} \)
97 \( 1 + 3.58e88T + 6.64e176T^{2} \)
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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

−13.81033209409634174375264388492, −13.22716652453080658041735609416, −10.72706581580217137107008922522, −9.592789556248329870816537767055, −8.364592346544513504668644354207, −6.07621847382744094365184216419, −5.21415148062023115787017345883, −3.03532685951216845715275839283, −1.39266514465652854674679254581, 0, 1.39266514465652854674679254581, 3.03532685951216845715275839283, 5.21415148062023115787017345883, 6.07621847382744094365184216419, 8.364592346544513504668644354207, 9.592789556248329870816537767055, 10.72706581580217137107008922522, 13.22716652453080658041735609416, 13.81033209409634174375264388492

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