Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 55
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.53e8·2-s − 1.66e12·3-s + 2.82e16·4-s + 1.07e19·5-s + 4.23e20·6-s − 1.10e23·7-s + 1.97e24·8-s − 1.71e26·9-s − 2.71e27·10-s − 7.94e28·11-s − 4.71e28·12-s + 3.15e30·13-s + 2.79e31·14-s − 1.79e31·15-s − 1.51e33·16-s + 3.96e33·17-s + 4.35e34·18-s + 1.06e35·19-s + 3.02e35·20-s + 1.84e35·21-s + 2.01e37·22-s + 2.36e37·23-s − 3.29e36·24-s − 1.62e38·25-s − 8.00e38·26-s + 5.77e38·27-s − 3.11e39·28-s + ⋯
L(s)  = 1  − 1.33·2-s − 0.126·3-s + 0.783·4-s + 0.643·5-s + 0.168·6-s − 0.634·7-s + 0.288·8-s − 0.984·9-s − 0.859·10-s − 1.82·11-s − 0.0990·12-s + 0.734·13-s + 0.847·14-s − 0.0813·15-s − 1.16·16-s + 0.576·17-s + 1.31·18-s + 0.729·19-s + 0.504·20-s + 0.0802·21-s + 2.43·22-s + 0.844·23-s − 0.0364·24-s − 0.585·25-s − 0.981·26-s + 0.250·27-s − 0.497·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(56-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+55/2) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(55\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :55/2),\ 1)$
$L(28)$  $\approx$  $0.657403$
$L(\frac12)$  $\approx$  $0.657403$
$L(\frac{57}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, \(F_p\) is a polynomial of degree 2.
$p$$F_p$
good2 \( 1 + 2.53e8T + 3.60e16T^{2} \)
3 \( 1 + 1.66e12T + 1.74e26T^{2} \)
5 \( 1 - 1.07e19T + 2.77e38T^{2} \)
7 \( 1 + 1.10e23T + 3.02e46T^{2} \)
11 \( 1 + 7.94e28T + 1.89e57T^{2} \)
13 \( 1 - 3.15e30T + 1.84e61T^{2} \)
17 \( 1 - 3.96e33T + 4.72e67T^{2} \)
19 \( 1 - 1.06e35T + 2.14e70T^{2} \)
23 \( 1 - 2.36e37T + 7.85e74T^{2} \)
29 \( 1 + 2.07e40T + 2.70e80T^{2} \)
31 \( 1 - 1.26e41T + 1.05e82T^{2} \)
37 \( 1 - 1.05e43T + 1.78e86T^{2} \)
41 \( 1 - 3.96e44T + 5.04e88T^{2} \)
43 \( 1 - 4.12e44T + 6.93e89T^{2} \)
47 \( 1 - 1.78e46T + 9.23e91T^{2} \)
53 \( 1 + 1.03e47T + 6.84e94T^{2} \)
59 \( 1 - 3.49e48T + 2.49e97T^{2} \)
61 \( 1 + 1.91e48T + 1.56e98T^{2} \)
67 \( 1 - 1.84e49T + 2.71e100T^{2} \)
71 \( 1 - 7.05e50T + 6.59e101T^{2} \)
73 \( 1 + 2.58e51T + 3.03e102T^{2} \)
79 \( 1 - 9.12e51T + 2.34e104T^{2} \)
83 \( 1 - 4.34e52T + 3.54e105T^{2} \)
89 \( 1 - 4.67e53T + 1.64e107T^{2} \)
97 \( 1 + 8.39e53T + 1.87e109T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.86303680419148520095675615549, −17.59401341441945158428791142920, −16.10315799827423832519098423060, −13.41925023132952489917618904491, −10.81170957746528457970535173058, −9.438438160703926861457700939837, −7.84000730608031809753268719439, −5.68326504619685570106718169026, −2.62468760178042997638962466692, −0.67796267284066736667574685265, 0.67796267284066736667574685265, 2.62468760178042997638962466692, 5.68326504619685570106718169026, 7.84000730608031809753268719439, 9.438438160703926861457700939837, 10.81170957746528457970535173058, 13.41925023132952489917618904491, 16.10315799827423832519098423060, 17.59401341441945158428791142920, 18.86303680419148520095675615549

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