Properties

Label 2-1-1.1-c53-0-3
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  + 1.29e8·2-s + 1.34e12·3-s + 7.73e15·4-s − 4.81e18·5-s + 1.73e20·6-s − 1.66e22·7-s − 1.64e23·8-s − 1.75e25·9-s − 6.22e26·10-s − 9.74e26·11-s + 1.03e28·12-s + 4.82e29·13-s − 2.14e30·14-s − 6.47e30·15-s − 9.09e31·16-s + 5.02e32·17-s − 2.27e33·18-s − 1.38e34·19-s − 3.72e34·20-s − 2.23e34·21-s − 1.26e35·22-s − 5.29e35·23-s − 2.21e35·24-s + 1.20e37·25-s + 6.23e37·26-s − 4.96e37·27-s − 1.28e38·28-s + ⋯
L(s)  = 1  + 1.36·2-s + 0.305·3-s + 0.858·4-s − 1.44·5-s + 0.416·6-s − 0.668·7-s − 0.192·8-s − 0.906·9-s − 1.96·10-s − 0.246·11-s + 0.262·12-s + 1.45·13-s − 0.911·14-s − 0.441·15-s − 1.12·16-s + 1.24·17-s − 1.23·18-s − 1.79·19-s − 1.24·20-s − 0.204·21-s − 0.335·22-s − 0.434·23-s − 0.0588·24-s + 1.08·25-s + 1.98·26-s − 0.582·27-s − 0.574·28-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 \( 1 - 1.29e8T + 9.00e15T^{2} \)
3 \( 1 - 1.34e12T + 1.93e25T^{2} \)
5 \( 1 + 4.81e18T + 1.11e37T^{2} \)
7 \( 1 + 1.66e22T + 6.16e44T^{2} \)
11 \( 1 + 9.74e26T + 1.56e55T^{2} \)
13 \( 1 - 4.82e29T + 1.09e59T^{2} \)
17 \( 1 - 5.02e32T + 1.63e65T^{2} \)
19 \( 1 + 1.38e34T + 5.94e67T^{2} \)
23 \( 1 + 5.29e35T + 1.48e72T^{2} \)
29 \( 1 + 3.32e38T + 3.21e77T^{2} \)
31 \( 1 - 2.69e39T + 1.10e79T^{2} \)
37 \( 1 - 2.08e41T + 1.30e83T^{2} \)
41 \( 1 - 2.96e42T + 3.00e85T^{2} \)
43 \( 1 - 1.06e43T + 3.74e86T^{2} \)
47 \( 1 + 2.94e44T + 4.18e88T^{2} \)
53 \( 1 + 5.52e45T + 2.43e91T^{2} \)
59 \( 1 - 1.07e47T + 7.16e93T^{2} \)
61 \( 1 - 3.62e46T + 4.19e94T^{2} \)
67 \( 1 - 7.15e47T + 6.05e96T^{2} \)
71 \( 1 + 9.79e48T + 1.30e98T^{2} \)
73 \( 1 + 2.48e49T + 5.70e98T^{2} \)
79 \( 1 + 2.29e50T + 3.75e100T^{2} \)
83 \( 1 + 8.70e50T + 5.14e101T^{2} \)
89 \( 1 - 2.82e51T + 2.07e103T^{2} \)
97 \( 1 + 5.70e52T + 1.99e105T^{2} \)
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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

−19.19351352257113457532888275147, −15.99812554526473854969102817702, −14.64201889273208358189579194485, −12.90052040179762709374073777583, −11.41715500917187292499776701244, −8.327112005454431297516498750343, −6.08708573159931613225512413659, −4.05411492846685872317216066274, −3.10089646196511199445192809885, 0, 3.10089646196511199445192809885, 4.05411492846685872317216066274, 6.08708573159931613225512413659, 8.327112005454431297516498750343, 11.41715500917187292499776701244, 12.90052040179762709374073777583, 14.64201889273208358189579194485, 15.99812554526473854969102817702, 19.19351352257113457532888275147

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