Properties

Label 2-1-1.1-c83-0-3
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  − 3.76e12·2-s − 6.68e19·3-s + 4.48e24·4-s + 1.95e28·5-s + 2.51e32·6-s + 1.66e35·7-s + 1.95e37·8-s + 4.81e38·9-s − 7.35e40·10-s + 1.57e43·11-s − 2.99e44·12-s + 1.54e46·13-s − 6.25e47·14-s − 1.30e48·15-s − 1.16e50·16-s − 2.86e50·17-s − 1.80e51·18-s + 1.03e53·19-s + 8.76e52·20-s − 1.11e55·21-s − 5.93e55·22-s − 2.08e56·23-s − 1.30e57·24-s − 9.95e57·25-s − 5.80e58·26-s + 2.34e59·27-s + 7.44e59·28-s + ⋯
L(s)  = 1  − 1.20·2-s − 1.05·3-s + 0.463·4-s + 0.192·5-s + 1.28·6-s + 1.40·7-s + 0.649·8-s + 0.120·9-s − 0.232·10-s + 0.955·11-s − 0.490·12-s + 0.911·13-s − 1.70·14-s − 0.203·15-s − 1.24·16-s − 0.247·17-s − 0.145·18-s + 0.884·19-s + 0.0891·20-s − 1.49·21-s − 1.15·22-s − 0.642·23-s − 0.687·24-s − 0.963·25-s − 1.10·26-s + 0.930·27-s + 0.653·28-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

\(L(42)\) \(\approx\) \(0.9015721133\)
\(L(\frac12)\) \(\approx\) \(0.9015721133\)
\(L(\frac{85}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 3.76e12T + 9.67e24T^{2} \)
3 \( 1 + 6.68e19T + 3.99e39T^{2} \)
5 \( 1 - 1.95e28T + 1.03e58T^{2} \)
7 \( 1 - 1.66e35T + 1.39e70T^{2} \)
11 \( 1 - 1.57e43T + 2.72e86T^{2} \)
13 \( 1 - 1.54e46T + 2.86e92T^{2} \)
17 \( 1 + 2.86e50T + 1.34e102T^{2} \)
19 \( 1 - 1.03e53T + 1.36e106T^{2} \)
23 \( 1 + 2.08e56T + 1.05e113T^{2} \)
29 \( 1 - 6.67e60T + 2.39e121T^{2} \)
31 \( 1 + 1.54e62T + 6.06e123T^{2} \)
37 \( 1 - 2.28e65T + 1.44e130T^{2} \)
41 \( 1 - 3.48e66T + 7.26e133T^{2} \)
43 \( 1 - 4.05e67T + 3.78e135T^{2} \)
47 \( 1 - 2.43e69T + 6.08e138T^{2} \)
53 \( 1 + 3.24e71T + 1.30e143T^{2} \)
59 \( 1 + 3.03e73T + 9.56e146T^{2} \)
61 \( 1 + 6.07e73T + 1.52e148T^{2} \)
67 \( 1 + 5.36e75T + 3.66e151T^{2} \)
71 \( 1 + 8.80e76T + 4.51e153T^{2} \)
73 \( 1 + 1.58e76T + 4.52e154T^{2} \)
79 \( 1 - 9.61e77T + 3.18e157T^{2} \)
83 \( 1 - 6.23e79T + 1.92e159T^{2} \)
89 \( 1 - 3.80e80T + 6.30e161T^{2} \)
97 \( 1 + 8.03e81T + 7.98e164T^{2} \)
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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

−16.34499961809415828026952923808, −14.08735111126821332388550334482, −11.65324184055942384735767582904, −10.80263429967867498850506587561, −9.099911642198438798651423212637, −7.74218041722544439697493183072, −5.96350173437053453213769990248, −4.44693571510118527574394580946, −1.64981835493912290174345703995, −0.76286909682885747196394090707, 0.76286909682885747196394090707, 1.64981835493912290174345703995, 4.44693571510118527574394580946, 5.96350173437053453213769990248, 7.74218041722544439697493183072, 9.099911642198438798651423212637, 10.80263429967867498850506587561, 11.65324184055942384735767582904, 14.08735111126821332388550334482, 16.34499961809415828026952923808

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