Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 105
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.93e15·2-s + 1.55e25·3-s − 5.33e30·4-s + 1.46e36·5-s + 9.23e40·6-s + 8.76e43·7-s − 2.72e47·8-s + 1.16e50·9-s + 8.68e51·10-s − 7.28e54·11-s − 8.29e55·12-s − 4.19e58·13-s + 5.20e59·14-s + 2.27e61·15-s − 1.40e63·16-s + 8.65e63·17-s + 6.92e65·18-s − 9.21e66·19-s − 7.79e66·20-s + 1.36e69·21-s − 4.32e70·22-s − 1.62e69·23-s − 4.23e72·24-s − 2.25e73·25-s − 2.48e74·26-s − 1.32e74·27-s − 4.67e74·28-s + ⋯
L(s)  = 1  + 0.931·2-s + 1.38·3-s − 0.131·4-s + 0.294·5-s + 1.29·6-s + 0.375·7-s − 1.05·8-s + 0.931·9-s + 0.274·10-s − 1.54·11-s − 0.182·12-s − 1.38·13-s + 0.350·14-s + 0.409·15-s − 0.851·16-s + 0.218·17-s + 0.868·18-s − 0.675·19-s − 0.0387·20-s + 0.522·21-s − 1.44·22-s − 0.00523·23-s − 1.46·24-s − 0.913·25-s − 1.28·26-s − 0.0948·27-s − 0.0494·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(105\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :105/2),\ -1)\)
\(L(53)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{107}{2})\)   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 \( 1 - 5.93e15T + 4.05e31T^{2} \)
3 \( 1 - 1.55e25T + 1.25e50T^{2} \)
5 \( 1 - 1.46e36T + 2.46e73T^{2} \)
7 \( 1 - 8.76e43T + 5.43e88T^{2} \)
11 \( 1 + 7.28e54T + 2.21e109T^{2} \)
13 \( 1 + 4.19e58T + 9.20e116T^{2} \)
17 \( 1 - 8.65e63T + 1.57e129T^{2} \)
19 \( 1 + 9.21e66T + 1.85e134T^{2} \)
23 \( 1 + 1.62e69T + 9.58e142T^{2} \)
29 \( 1 - 9.56e76T + 3.56e153T^{2} \)
31 \( 1 + 1.52e78T + 3.91e156T^{2} \)
37 \( 1 + 2.78e82T + 4.58e164T^{2} \)
41 \( 1 - 5.59e84T + 2.19e169T^{2} \)
43 \( 1 - 2.87e84T + 3.26e171T^{2} \)
47 \( 1 - 8.04e87T + 3.71e175T^{2} \)
53 \( 1 - 5.94e90T + 1.11e181T^{2} \)
59 \( 1 + 1.24e93T + 8.69e185T^{2} \)
61 \( 1 - 3.83e93T + 2.88e187T^{2} \)
67 \( 1 + 7.25e95T + 5.46e191T^{2} \)
71 \( 1 + 3.02e96T + 2.41e194T^{2} \)
73 \( 1 + 1.13e98T + 4.45e195T^{2} \)
79 \( 1 - 1.66e99T + 1.78e199T^{2} \)
83 \( 1 + 3.34e100T + 3.18e201T^{2} \)
89 \( 1 - 3.73e101T + 4.85e204T^{2} \)
97 \( 1 + 2.81e104T + 4.08e208T^{2} \)
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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

−13.54589737561230241139805621595, −12.41016285801842011064891642272, −10.10611960649982188021238843889, −8.748400818219853857012357048878, −7.56903182565964511776826582008, −5.49219088031924567290155405577, −4.35698957308761850995176144597, −2.92821552560790134755792987695, −2.22584091773723128503876906417, 0, 2.22584091773723128503876906417, 2.92821552560790134755792987695, 4.35698957308761850995176144597, 5.49219088031924567290155405577, 7.56903182565964511776826582008, 8.748400818219853857012357048878, 10.10611960649982188021238843889, 12.41016285801842011064891642272, 13.54589737561230241139805621595

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