Properties

Label 2-1-1.1-c83-0-0
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $43.6272$
Root an. cond. $6.60508$
Motivic weight $83$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.82e12·2-s + 1.57e19·3-s − 6.34e24·4-s − 2.84e28·5-s − 2.86e31·6-s − 2.18e35·7-s + 2.92e37·8-s − 3.74e39·9-s + 5.18e40·10-s − 1.21e43·11-s − 9.97e43·12-s − 1.12e46·13-s + 3.97e47·14-s − 4.47e47·15-s + 8.18e48·16-s − 8.27e49·17-s + 6.82e51·18-s − 1.06e53·19-s + 1.80e53·20-s − 3.42e54·21-s + 2.21e55·22-s − 5.45e56·23-s + 4.58e56·24-s − 9.52e57·25-s + 2.04e58·26-s − 1.21e59·27-s + 1.38e60·28-s + ⋯
L(s)  = 1  − 0.586·2-s + 0.248·3-s − 0.656·4-s − 0.279·5-s − 0.145·6-s − 1.85·7-s + 0.970·8-s − 0.938·9-s + 0.164·10-s − 0.737·11-s − 0.163·12-s − 0.663·13-s + 1.08·14-s − 0.0695·15-s + 0.0874·16-s − 0.0714·17-s + 0.549·18-s − 0.910·19-s + 0.183·20-s − 0.460·21-s + 0.432·22-s − 1.68·23-s + 0.241·24-s − 0.921·25-s + 0.388·26-s − 0.482·27-s + 1.21·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(84-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+83/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(43.6272\)
Root analytic conductor: \(6.60508\)
Motivic weight: \(83\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :83/2),\ 1)\)

Particular Values

\(L(42)\) \(\approx\) \(0.02894907561\)
\(L(\frac12)\) \(\approx\) \(0.02894907561\)
\(L(\frac{85}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.82e12T + 9.67e24T^{2} \)
3 \( 1 - 1.57e19T + 3.99e39T^{2} \)
5 \( 1 + 2.84e28T + 1.03e58T^{2} \)
7 \( 1 + 2.18e35T + 1.39e70T^{2} \)
11 \( 1 + 1.21e43T + 2.72e86T^{2} \)
13 \( 1 + 1.12e46T + 2.86e92T^{2} \)
17 \( 1 + 8.27e49T + 1.34e102T^{2} \)
19 \( 1 + 1.06e53T + 1.36e106T^{2} \)
23 \( 1 + 5.45e56T + 1.05e113T^{2} \)
29 \( 1 + 1.24e60T + 2.39e121T^{2} \)
31 \( 1 - 4.29e61T + 6.06e123T^{2} \)
37 \( 1 + 8.01e64T + 1.44e130T^{2} \)
41 \( 1 - 7.03e66T + 7.26e133T^{2} \)
43 \( 1 - 8.10e67T + 3.78e135T^{2} \)
47 \( 1 - 2.80e69T + 6.08e138T^{2} \)
53 \( 1 + 3.32e71T + 1.30e143T^{2} \)
59 \( 1 + 5.84e73T + 9.56e146T^{2} \)
61 \( 1 - 6.63e73T + 1.52e148T^{2} \)
67 \( 1 - 2.26e75T + 3.66e151T^{2} \)
71 \( 1 + 8.10e75T + 4.51e153T^{2} \)
73 \( 1 + 2.53e77T + 4.52e154T^{2} \)
79 \( 1 - 1.36e78T + 3.18e157T^{2} \)
83 \( 1 + 5.99e79T + 1.92e159T^{2} \)
89 \( 1 - 5.14e80T + 6.30e161T^{2} \)
97 \( 1 + 2.08e82T + 7.98e164T^{2} \)
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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

−15.87243098175375988233679842058, −13.87164102847321485729916103403, −12.54410943955747686832196391966, −10.26613113484737005829444572786, −9.175851008958521833914342319062, −7.78749232112316774677119278443, −5.93169888106350310983940573615, −3.97008373389652681899794330110, −2.54919841922959855902120281457, −0.098883314690710113132541806307, 0.098883314690710113132541806307, 2.54919841922959855902120281457, 3.97008373389652681899794330110, 5.93169888106350310983940573615, 7.78749232112316774677119278443, 9.175851008958521833914342319062, 10.26613113484737005829444572786, 12.54410943955747686832196391966, 13.87164102847321485729916103403, 15.87243098175375988233679842058

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