Properties

Label 2-1-1.1-c53-0-0
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $17.7903$
Root an. cond. $4.21785$
Motivic weight $53$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.38e8·2-s − 5.95e12·3-s + 1.01e16·4-s − 5.35e18·5-s + 8.25e20·6-s + 2.55e22·7-s − 1.64e23·8-s + 1.60e25·9-s + 7.42e26·10-s + 2.25e27·11-s − 6.06e28·12-s + 1.44e29·13-s − 3.53e30·14-s + 3.19e31·15-s − 6.90e31·16-s − 6.08e32·17-s − 2.22e33·18-s + 5.84e33·19-s − 5.46e34·20-s − 1.52e35·21-s − 3.12e35·22-s + 9.54e35·23-s + 9.79e35·24-s + 1.76e37·25-s − 2.00e37·26-s + 1.97e37·27-s + 2.60e38·28-s + ⋯
L(s)  = 1  − 1.46·2-s − 1.35·3-s + 1.13·4-s − 1.60·5-s + 1.97·6-s + 1.02·7-s − 0.192·8-s + 0.828·9-s + 2.34·10-s + 0.571·11-s − 1.53·12-s + 0.436·13-s − 1.50·14-s + 2.17·15-s − 0.850·16-s − 1.50·17-s − 1.21·18-s + 0.757·19-s − 1.82·20-s − 1.39·21-s − 0.833·22-s + 0.783·23-s + 0.260·24-s + 1.58·25-s − 0.637·26-s + 0.231·27-s + 1.16·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(54-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+53/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(17.7903\)
Root analytic conductor: \(4.21785\)
Motivic weight: \(53\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :53/2),\ -1)\)

Particular Values

\(L(27)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{55}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.38e8T + 9.00e15T^{2} \)
3 \( 1 + 5.95e12T + 1.93e25T^{2} \)
5 \( 1 + 5.35e18T + 1.11e37T^{2} \)
7 \( 1 - 2.55e22T + 6.16e44T^{2} \)
11 \( 1 - 2.25e27T + 1.56e55T^{2} \)
13 \( 1 - 1.44e29T + 1.09e59T^{2} \)
17 \( 1 + 6.08e32T + 1.63e65T^{2} \)
19 \( 1 - 5.84e33T + 5.94e67T^{2} \)
23 \( 1 - 9.54e35T + 1.48e72T^{2} \)
29 \( 1 + 4.38e37T + 3.21e77T^{2} \)
31 \( 1 - 8.43e38T + 1.10e79T^{2} \)
37 \( 1 - 1.19e41T + 1.30e83T^{2} \)
41 \( 1 + 8.08e42T + 3.00e85T^{2} \)
43 \( 1 - 1.66e43T + 3.74e86T^{2} \)
47 \( 1 + 7.16e42T + 4.18e88T^{2} \)
53 \( 1 - 3.19e45T + 2.43e91T^{2} \)
59 \( 1 - 3.07e46T + 7.16e93T^{2} \)
61 \( 1 + 2.46e47T + 4.19e94T^{2} \)
67 \( 1 - 1.37e48T + 6.05e96T^{2} \)
71 \( 1 - 1.40e49T + 1.30e98T^{2} \)
73 \( 1 + 1.59e49T + 5.70e98T^{2} \)
79 \( 1 + 1.73e50T + 3.75e100T^{2} \)
83 \( 1 + 1.32e51T + 5.14e101T^{2} \)
89 \( 1 + 6.74e51T + 2.07e103T^{2} \)
97 \( 1 - 4.44e52T + 1.99e105T^{2} \)
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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

−18.33694812547888774415463563067, −17.09998277905143017185942393996, −15.69958258626181350212066838056, −11.60301157721301318006491683168, −11.02693080545101520574031167217, −8.518912255990319196151859506115, −7.04990757063851416267034010110, −4.54848879576851445901379130219, −1.11128581718251547380353913934, 0, 1.11128581718251547380353913934, 4.54848879576851445901379130219, 7.04990757063851416267034010110, 8.518912255990319196151859506115, 11.02693080545101520574031167217, 11.60301157721301318006491683168, 15.69958258626181350212066838056, 17.09998277905143017185942393996, 18.33694812547888774415463563067

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