Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 73
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.14e10·2-s − 1.08e17·3-s − 4.34e21·4-s − 5.37e25·5-s + 7.78e27·6-s + 1.02e31·7-s + 9.84e32·8-s − 5.57e34·9-s + 3.83e36·10-s − 4.39e37·11-s + 4.73e38·12-s + 5.53e40·13-s − 7.31e41·14-s + 5.85e42·15-s − 2.93e43·16-s + 9.40e44·17-s + 3.97e45·18-s + 1.72e46·19-s + 2.33e47·20-s − 1.11e48·21-s + 3.13e48·22-s − 5.77e49·23-s − 1.07e50·24-s + 1.82e51·25-s − 3.95e51·26-s + 1.34e52·27-s − 4.44e52·28-s + ⋯
L(s)  = 1  − 0.734·2-s − 0.419·3-s − 0.459·4-s − 1.65·5-s + 0.308·6-s + 1.46·7-s + 1.07·8-s − 0.824·9-s + 1.21·10-s − 0.428·11-s + 0.192·12-s + 1.21·13-s − 1.07·14-s + 0.692·15-s − 0.328·16-s + 1.15·17-s + 0.605·18-s + 0.365·19-s + 0.759·20-s − 0.612·21-s + 0.314·22-s − 1.14·23-s − 0.449·24-s + 1.72·25-s − 0.892·26-s + 0.764·27-s − 0.671·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(74-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+73/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(73\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :73/2),\ -1)\)
\(L(37)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{75}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 2.
$p$$F_p(T)$
good2 \( 1 + 7.14e10T + 9.44e21T^{2} \)
3 \( 1 + 1.08e17T + 6.75e34T^{2} \)
5 \( 1 + 5.37e25T + 1.05e51T^{2} \)
7 \( 1 - 1.02e31T + 4.92e61T^{2} \)
11 \( 1 + 4.39e37T + 1.05e76T^{2} \)
13 \( 1 - 5.53e40T + 2.07e81T^{2} \)
17 \( 1 - 9.40e44T + 6.64e89T^{2} \)
19 \( 1 - 1.72e46T + 2.23e93T^{2} \)
23 \( 1 + 5.77e49T + 2.54e99T^{2} \)
29 \( 1 + 8.69e52T + 5.68e106T^{2} \)
31 \( 1 - 1.48e54T + 7.40e108T^{2} \)
37 \( 1 + 2.64e57T + 3.01e114T^{2} \)
41 \( 1 - 3.07e57T + 5.41e117T^{2} \)
43 \( 1 - 3.24e59T + 1.75e119T^{2} \)
47 \( 1 + 8.51e60T + 1.15e122T^{2} \)
53 \( 1 + 8.36e62T + 7.44e125T^{2} \)
59 \( 1 - 2.35e64T + 1.87e129T^{2} \)
61 \( 1 - 5.89e64T + 2.13e130T^{2} \)
67 \( 1 - 6.87e66T + 2.01e133T^{2} \)
71 \( 1 - 8.24e66T + 1.38e135T^{2} \)
73 \( 1 + 1.53e68T + 1.05e136T^{2} \)
79 \( 1 + 3.64e68T + 3.36e138T^{2} \)
83 \( 1 - 1.69e70T + 1.23e140T^{2} \)
89 \( 1 - 7.76e69T + 2.02e142T^{2} \)
97 \( 1 - 4.51e71T + 1.08e145T^{2} \)
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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

−16.08862940295662619546528687624, −14.26776549475682219848603121287, −11.83359689965210690315775722220, −10.82132371658483699935709297657, −8.391836326963413296838111280111, −7.85227310763809429256627856750, −5.16327254886260277742658361087, −3.78210053216481678964923402046, −1.16525561455138451385061388305, 0, 1.16525561455138451385061388305, 3.78210053216481678964923402046, 5.16327254886260277742658361087, 7.85227310763809429256627856750, 8.391836326963413296838111280111, 10.82132371658483699935709297657, 11.83359689965210690315775722220, 14.26776549475682219848603121287, 16.08862940295662619546528687624

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