Properties

Degree 2
Conductor $ 1 $
Sign $-1$
Motivic weight 93
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.35e14·2-s − 1.21e22·3-s + 8.34e27·4-s + 3.52e32·5-s + 1.64e36·6-s − 7.46e38·7-s + 2.10e41·8-s − 8.75e43·9-s − 4.75e46·10-s − 9.09e47·11-s − 1.01e50·12-s − 6.25e51·13-s + 1.00e53·14-s − 4.28e54·15-s − 1.11e56·16-s + 2.83e57·17-s + 1.18e58·18-s − 8.20e57·19-s + 2.94e60·20-s + 9.08e60·21-s + 1.22e62·22-s − 2.05e63·23-s − 2.55e63·24-s + 2.30e64·25-s + 8.44e65·26-s + 3.93e66·27-s − 6.23e66·28-s + ⋯
L(s)  = 1  − 1.35·2-s − 0.792·3-s + 0.842·4-s + 1.10·5-s + 1.07·6-s − 0.376·7-s + 0.213·8-s − 0.371·9-s − 1.50·10-s − 0.342·11-s − 0.668·12-s − 0.994·13-s + 0.511·14-s − 0.878·15-s − 1.13·16-s + 1.72·17-s + 0.504·18-s − 0.0283·19-s + 0.934·20-s + 0.298·21-s + 0.464·22-s − 0.984·23-s − 0.169·24-s + 0.228·25-s + 1.35·26-s + 1.08·27-s − 0.317·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(94-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+93/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(93\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1,\ (\ :93/2),\ -1)\)
\(L(47)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{95}{2})\)   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 \( 1 + 1.35e14T + 9.90e27T^{2} \)
3 \( 1 + 1.21e22T + 2.35e44T^{2} \)
5 \( 1 - 3.52e32T + 1.00e65T^{2} \)
7 \( 1 + 7.46e38T + 3.92e78T^{2} \)
11 \( 1 + 9.09e47T + 7.07e96T^{2} \)
13 \( 1 + 6.25e51T + 3.95e103T^{2} \)
17 \( 1 - 2.83e57T + 2.70e114T^{2} \)
19 \( 1 + 8.20e57T + 8.39e118T^{2} \)
23 \( 1 + 2.05e63T + 4.37e126T^{2} \)
29 \( 1 - 1.94e68T + 1.00e136T^{2} \)
31 \( 1 - 8.59e68T + 4.97e138T^{2} \)
37 \( 1 - 9.63e71T + 6.96e145T^{2} \)
41 \( 1 - 1.11e75T + 9.74e149T^{2} \)
43 \( 1 + 1.49e76T + 8.17e151T^{2} \)
47 \( 1 + 6.74e75T + 3.19e155T^{2} \)
53 \( 1 - 2.18e80T + 2.27e160T^{2} \)
59 \( 1 + 4.18e82T + 4.88e164T^{2} \)
61 \( 1 - 3.83e82T + 1.08e166T^{2} \)
67 \( 1 - 2.76e84T + 6.68e169T^{2} \)
71 \( 1 - 1.96e86T + 1.46e172T^{2} \)
73 \( 1 + 1.54e86T + 1.94e173T^{2} \)
79 \( 1 - 8.28e87T + 3.01e176T^{2} \)
83 \( 1 - 1.06e89T + 2.98e178T^{2} \)
89 \( 1 + 1.04e90T + 1.96e181T^{2} \)
97 \( 1 + 3.92e92T + 5.88e184T^{2} \)
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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

−14.01405371683185925079271383618, −12.10148178572819624359717517584, −10.33430700183150956669022495641, −9.698765238117515345759443998310, −8.036956303869130759088384402349, −6.40143741803960586364223793336, −5.15604176592037110145585460923, −2.56495704095559300976348074634, −1.12993969144782936120472009162, 0, 1.12993969144782936120472009162, 2.56495704095559300976348074634, 5.15604176592037110145585460923, 6.40143741803960586364223793336, 8.036956303869130759088384402349, 9.698765238117515345759443998310, 10.33430700183150956669022495641, 12.10148178572819624359717517584, 14.01405371683185925079271383618

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