Properties

Degree 2
Conductor $ 1 $
Sign $1$
Motivic weight 59
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.42e9·2-s − 1.64e14·3-s + 1.44e18·4-s + 2.46e20·5-s + 2.33e23·6-s − 6.31e24·7-s − 1.23e27·8-s + 1.28e28·9-s − 3.49e29·10-s − 1.85e29·11-s − 2.37e32·12-s − 9.37e32·13-s + 8.97e33·14-s − 4.03e34·15-s + 9.23e35·16-s − 2.12e36·17-s − 1.82e37·18-s + 2.93e37·19-s + 3.55e38·20-s + 1.03e39·21-s + 2.63e38·22-s − 1.73e40·23-s + 2.02e41·24-s − 1.12e41·25-s + 1.33e42·26-s + 2.12e41·27-s − 9.12e42·28-s + ⋯
L(s)  = 1  − 1.87·2-s − 1.38·3-s + 2.50·4-s + 0.590·5-s + 2.58·6-s − 0.740·7-s − 2.82·8-s + 0.908·9-s − 1.10·10-s − 0.0352·11-s − 3.46·12-s − 1.29·13-s + 1.38·14-s − 0.815·15-s + 2.77·16-s − 1.07·17-s − 1.70·18-s + 0.555·19-s + 1.48·20-s + 1.02·21-s + 0.0659·22-s − 1.17·23-s + 3.89·24-s − 0.651·25-s + 2.41·26-s + 0.126·27-s − 1.85·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \,\Lambda(60-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+59/2) \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(59\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1,\ (\ :59/2),\ 1)$
$L(30)$  $\approx$  $0.180619$
$L(\frac12)$  $\approx$  $0.180619$
$L(\frac{61}{2})$   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 \( 1 + 1.42e9T + 5.76e17T^{2} \)
3 \( 1 + 1.64e14T + 1.41e28T^{2} \)
5 \( 1 - 2.46e20T + 1.73e41T^{2} \)
7 \( 1 + 6.31e24T + 7.25e49T^{2} \)
11 \( 1 + 1.85e29T + 2.76e61T^{2} \)
13 \( 1 + 9.37e32T + 5.28e65T^{2} \)
17 \( 1 + 2.12e36T + 3.94e72T^{2} \)
19 \( 1 - 2.93e37T + 2.79e75T^{2} \)
23 \( 1 + 1.73e40T + 2.19e80T^{2} \)
29 \( 1 - 2.89e41T + 1.91e86T^{2} \)
31 \( 1 + 1.46e44T + 9.78e87T^{2} \)
37 \( 1 + 2.03e46T + 3.34e92T^{2} \)
41 \( 1 - 2.62e47T + 1.42e95T^{2} \)
43 \( 1 - 8.93e47T + 2.36e96T^{2} \)
47 \( 1 - 2.53e49T + 4.50e98T^{2} \)
53 \( 1 - 1.22e51T + 5.39e101T^{2} \)
59 \( 1 - 5.85e50T + 3.02e104T^{2} \)
61 \( 1 + 6.37e52T + 2.16e105T^{2} \)
67 \( 1 - 6.46e53T + 5.47e107T^{2} \)
71 \( 1 + 1.44e54T + 1.67e109T^{2} \)
73 \( 1 + 2.42e54T + 8.63e109T^{2} \)
79 \( 1 - 4.97e54T + 9.12e111T^{2} \)
83 \( 1 - 3.44e56T + 1.68e113T^{2} \)
89 \( 1 - 3.85e57T + 1.03e115T^{2} \)
97 \( 1 + 2.95e58T + 1.65e117T^{2} \)
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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

−18.06289930536089275485428044744, −17.16741229993592399310771795827, −15.99694712740501186200387001381, −12.05525027679948517524853575776, −10.56136995219441419035718301415, −9.423065804145231908188850616699, −7.14358773262112345305516870612, −5.85938177333788607586037303447, −2.11862684631809459824583932265, −0.38173724720063210159473700199, 0.38173724720063210159473700199, 2.11862684631809459824583932265, 5.85938177333788607586037303447, 7.14358773262112345305516870612, 9.423065804145231908188850616699, 10.56136995219441419035718301415, 12.05525027679948517524853575776, 15.99694712740501186200387001381, 17.16741229993592399310771795827, 18.06289930536089275485428044744

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