Properties

Label 2-1-1.1-c53-0-2
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  − 8.51e7·2-s + 6.54e12·3-s − 1.75e15·4-s + 2.26e17·5-s − 5.56e20·6-s − 3.24e22·7-s + 9.16e23·8-s + 2.33e25·9-s − 1.92e25·10-s + 4.70e27·11-s − 1.15e28·12-s − 5.49e29·13-s + 2.76e30·14-s + 1.48e30·15-s − 6.21e31·16-s − 1.83e32·17-s − 1.99e33·18-s − 8.16e33·19-s − 3.98e32·20-s − 2.12e35·21-s − 4.00e35·22-s + 8.36e35·23-s + 5.99e36·24-s − 1.10e37·25-s + 4.68e37·26-s + 2.62e37·27-s + 5.71e37·28-s + ⋯
L(s)  = 1  − 0.897·2-s + 1.48·3-s − 0.195·4-s + 0.0679·5-s − 1.33·6-s − 1.30·7-s + 1.07·8-s + 1.20·9-s − 0.0609·10-s + 1.19·11-s − 0.290·12-s − 1.66·13-s + 1.17·14-s + 0.100·15-s − 0.766·16-s − 0.454·17-s − 1.08·18-s − 1.05·19-s − 0.0132·20-s − 1.94·21-s − 1.06·22-s + 0.686·23-s + 1.59·24-s − 0.995·25-s + 1.49·26-s + 0.307·27-s + 0.255·28-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 \( 1 + 8.51e7T + 9.00e15T^{2} \)
3 \( 1 - 6.54e12T + 1.93e25T^{2} \)
5 \( 1 - 2.26e17T + 1.11e37T^{2} \)
7 \( 1 + 3.24e22T + 6.16e44T^{2} \)
11 \( 1 - 4.70e27T + 1.56e55T^{2} \)
13 \( 1 + 5.49e29T + 1.09e59T^{2} \)
17 \( 1 + 1.83e32T + 1.63e65T^{2} \)
19 \( 1 + 8.16e33T + 5.94e67T^{2} \)
23 \( 1 - 8.36e35T + 1.48e72T^{2} \)
29 \( 1 + 7.80e38T + 3.21e77T^{2} \)
31 \( 1 + 3.24e39T + 1.10e79T^{2} \)
37 \( 1 + 2.65e40T + 1.30e83T^{2} \)
41 \( 1 - 5.60e41T + 3.00e85T^{2} \)
43 \( 1 - 2.91e42T + 3.74e86T^{2} \)
47 \( 1 + 3.43e43T + 4.18e88T^{2} \)
53 \( 1 - 5.37e45T + 2.43e91T^{2} \)
59 \( 1 - 1.17e47T + 7.16e93T^{2} \)
61 \( 1 + 6.10e46T + 4.19e94T^{2} \)
67 \( 1 - 5.08e47T + 6.05e96T^{2} \)
71 \( 1 + 2.80e48T + 1.30e98T^{2} \)
73 \( 1 + 1.95e49T + 5.70e98T^{2} \)
79 \( 1 - 1.95e50T + 3.75e100T^{2} \)
83 \( 1 + 9.78e49T + 5.14e101T^{2} \)
89 \( 1 - 4.98e50T + 2.07e103T^{2} \)
97 \( 1 + 1.99e52T + 1.99e105T^{2} \)
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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

−19.24579676330335209117023816408, −16.94636473248664722001432196719, −14.71597501873281346212262048807, −13.09996580144090454415565086225, −9.727609437338406029370805436637, −8.982004386372900206391170042899, −7.22583473262764177733905317129, −3.86475122900726968429150086588, −2.12491545208015807400443058476, 0, 2.12491545208015807400443058476, 3.86475122900726968429150086588, 7.22583473262764177733905317129, 8.982004386372900206391170042899, 9.727609437338406029370805436637, 13.09996580144090454415565086225, 14.71597501873281346212262048807, 16.94636473248664722001432196719, 19.24579676330335209117023816408

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