Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 69
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.72e9·2-s − 4.13e16·3-s − 5.45e20·4-s − 6.57e23·5-s + 2.78e26·6-s + 1.09e29·7-s + 7.63e30·8-s + 8.78e32·9-s + 4.42e33·10-s + 1.33e36·11-s + 2.25e37·12-s − 4.12e38·13-s − 7.34e38·14-s + 2.72e40·15-s + 2.70e41·16-s − 6.74e41·17-s − 5.91e42·18-s + 4.81e43·19-s + 3.58e44·20-s − 4.51e45·21-s − 8.95e45·22-s + 1.69e47·23-s − 3.16e47·24-s − 1.26e48·25-s + 2.77e48·26-s − 1.84e48·27-s − 5.95e49·28-s + ⋯
L(s)  = 1  − 0.276·2-s − 1.43·3-s − 0.923·4-s − 0.505·5-s + 0.396·6-s + 0.762·7-s + 0.532·8-s + 1.05·9-s + 0.139·10-s + 1.57·11-s + 1.32·12-s − 1.52·13-s − 0.211·14-s + 0.723·15-s + 0.775·16-s − 0.238·17-s − 0.291·18-s + 0.367·19-s + 0.466·20-s − 1.09·21-s − 0.435·22-s + 1.77·23-s − 0.762·24-s − 0.744·25-s + 0.423·26-s − 0.0765·27-s − 0.704·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\,\Lambda(70-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\Gamma_{\C}(s+69/2) \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(69\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1,\ (\ :69/2),\ -1)$
$L(35)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{71}{2})$   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 \( 1 + 6.72e9T + 5.90e20T^{2} \)
3 \( 1 + 4.13e16T + 8.34e32T^{2} \)
5 \( 1 + 6.57e23T + 1.69e48T^{2} \)
7 \( 1 - 1.09e29T + 2.05e58T^{2} \)
11 \( 1 - 1.33e36T + 7.17e71T^{2} \)
13 \( 1 + 4.12e38T + 7.27e76T^{2} \)
17 \( 1 + 6.74e41T + 7.96e84T^{2} \)
19 \( 1 - 4.81e43T + 1.71e88T^{2} \)
23 \( 1 - 1.69e47T + 9.10e93T^{2} \)
29 \( 1 + 1.58e50T + 8.04e100T^{2} \)
31 \( 1 + 2.69e51T + 8.01e102T^{2} \)
37 \( 1 + 8.15e53T + 1.60e108T^{2} \)
41 \( 1 - 2.61e55T + 1.91e111T^{2} \)
43 \( 1 - 4.10e56T + 5.12e112T^{2} \)
47 \( 1 - 9.62e56T + 2.37e115T^{2} \)
53 \( 1 - 1.75e59T + 9.44e118T^{2} \)
59 \( 1 + 1.91e61T + 1.54e122T^{2} \)
61 \( 1 - 5.14e61T + 1.54e123T^{2} \)
67 \( 1 + 5.82e62T + 9.98e125T^{2} \)
71 \( 1 - 6.68e63T + 5.45e127T^{2} \)
73 \( 1 + 1.20e64T + 3.70e128T^{2} \)
79 \( 1 + 1.97e65T + 8.63e130T^{2} \)
83 \( 1 - 5.46e65T + 2.60e132T^{2} \)
89 \( 1 + 8.41e66T + 3.22e134T^{2} \)
97 \( 1 + 3.94e68T + 1.22e137T^{2} \)
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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

−16.93306469208496848227181945503, −14.56308797712115861929071930855, −12.31641200673431184236453809240, −11.13186324419509530680078508307, −9.281628834778413159500848067949, −7.26931435417719647869548790433, −5.31032359578194051714993233073, −4.24124945859646473349903099146, −1.16933579666952814635707694032, 0, 1.16933579666952814635707694032, 4.24124945859646473349903099146, 5.31032359578194051714993233073, 7.26931435417719647869548790433, 9.281628834778413159500848067949, 11.13186324419509530680078508307, 12.31641200673431184236453809240, 14.56308797712115861929071930855, 16.93306469208496848227181945503

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