Properties

Degree 2
Conductor 2
Sign $1$
Motivic weight 83
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.19e12·2-s − 7.32e17·3-s + 4.83e24·4-s − 1.23e29·5-s + 1.61e30·6-s + 6.61e34·7-s − 1.06e37·8-s − 3.99e39·9-s + 2.72e41·10-s − 2.35e43·11-s − 3.54e42·12-s + 2.17e46·13-s − 1.45e47·14-s + 9.06e46·15-s + 2.33e49·16-s − 2.06e51·17-s + 8.77e51·18-s − 1.54e53·19-s − 5.98e53·20-s − 4.84e52·21-s + 5.17e55·22-s + 3.09e56·23-s + 7.78e54·24-s + 4.96e57·25-s − 4.78e58·26-s + 5.84e57·27-s + 3.19e59·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0115·3-s + 0.5·4-s − 1.21·5-s + 0.00819·6-s + 0.561·7-s − 0.353·8-s − 0.999·9-s + 0.860·10-s − 1.42·11-s − 0.00579·12-s + 1.28·13-s − 0.396·14-s + 0.0141·15-s + 0.250·16-s − 1.77·17-s + 0.707·18-s − 1.31·19-s − 0.608·20-s − 0.00650·21-s + 1.00·22-s + 0.953·23-s + 0.00409·24-s + 0.480·25-s − 0.908·26-s + 0.0231·27-s + 0.280·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(84-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+83/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(83\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2,\ (\ :83/2),\ 1)$
$L(42)$  $\approx$  $0.2684754695$
$L(\frac12)$  $\approx$  $0.2684754695$
$L(\frac{85}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + 2.19e12T \)
good3 \( 1 + 7.32e17T + 3.99e39T^{2} \)
5 \( 1 + 1.23e29T + 1.03e58T^{2} \)
7 \( 1 - 6.61e34T + 1.39e70T^{2} \)
11 \( 1 + 2.35e43T + 2.72e86T^{2} \)
13 \( 1 - 2.17e46T + 2.86e92T^{2} \)
17 \( 1 + 2.06e51T + 1.34e102T^{2} \)
19 \( 1 + 1.54e53T + 1.36e106T^{2} \)
23 \( 1 - 3.09e56T + 1.05e113T^{2} \)
29 \( 1 + 4.67e59T + 2.39e121T^{2} \)
31 \( 1 + 1.70e61T + 6.06e123T^{2} \)
37 \( 1 + 9.03e64T + 1.44e130T^{2} \)
41 \( 1 + 8.21e66T + 7.26e133T^{2} \)
43 \( 1 + 5.19e67T + 3.78e135T^{2} \)
47 \( 1 + 3.72e69T + 6.08e138T^{2} \)
53 \( 1 + 4.87e71T + 1.30e143T^{2} \)
59 \( 1 - 2.59e73T + 9.56e146T^{2} \)
61 \( 1 + 2.95e73T + 1.52e148T^{2} \)
67 \( 1 - 9.82e75T + 3.66e151T^{2} \)
71 \( 1 + 2.74e76T + 4.51e153T^{2} \)
73 \( 1 - 1.97e77T + 4.52e154T^{2} \)
79 \( 1 - 2.44e78T + 3.18e157T^{2} \)
83 \( 1 + 4.50e79T + 1.92e159T^{2} \)
89 \( 1 - 1.00e81T + 6.30e161T^{2} \)
97 \( 1 - 2.37e82T + 7.98e164T^{2} \)
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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

−13.12488263780520650151726988858, −11.31778640454824613380333939006, −10.88167920773412221064390714451, −8.593948661947357553317514111734, −8.157539316703607536004586605043, −6.58905180427988984388776399401, −4.89231643953424577113814838626, −3.36477498500808209903509353717, −2.01410371192593249767918229774, −0.27143987742286290018096475355, 0.27143987742286290018096475355, 2.01410371192593249767918229774, 3.36477498500808209903509353717, 4.89231643953424577113814838626, 6.58905180427988984388776399401, 8.157539316703607536004586605043, 8.593948661947357553317514111734, 10.88167920773412221064390714451, 11.31778640454824613380333939006, 13.12488263780520650151726988858

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