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 − 1.03e20·3-s + 4.83e24·4-s − 4.29e28·5-s + 2.27e32·6-s − 1.14e35·7-s − 1.06e37·8-s + 6.67e39·9-s + 9.44e40·10-s + 1.30e43·11-s − 4.99e44·12-s − 2.19e46·13-s + 2.51e47·14-s + 4.43e48·15-s + 2.33e49·16-s + 1.57e51·17-s − 1.46e52·18-s − 1.27e53·19-s − 2.07e53·20-s + 1.18e55·21-s − 2.86e55·22-s + 2.00e56·23-s + 1.09e57·24-s − 8.49e57·25-s + 4.83e58·26-s − 2.77e59·27-s − 5.52e59·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.63·3-s + 0.5·4-s − 0.422·5-s + 1.15·6-s − 0.968·7-s − 0.353·8-s + 1.67·9-s + 0.298·10-s + 0.787·11-s − 0.817·12-s − 1.29·13-s + 0.685·14-s + 0.690·15-s + 0.250·16-s + 1.36·17-s − 1.18·18-s − 1.08·19-s − 0.211·20-s + 1.58·21-s − 0.557·22-s + 0.616·23-s + 0.578·24-s − 0.821·25-s + 0.917·26-s − 1.10·27-s − 0.484·28-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  =  $(2,\ 2,\ (\ :83/2),\ 1)$
$L(42)$  $\approx$  $0.03413206805$
$L(\frac12)$  $\approx$  $0.03413206805$
$L(\frac{85}{2})$   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 \( 1 + 1.03e20T + 3.99e39T^{2} \)
5 \( 1 + 4.29e28T + 1.03e58T^{2} \)
7 \( 1 + 1.14e35T + 1.39e70T^{2} \)
11 \( 1 - 1.30e43T + 2.72e86T^{2} \)
13 \( 1 + 2.19e46T + 2.86e92T^{2} \)
17 \( 1 - 1.57e51T + 1.34e102T^{2} \)
19 \( 1 + 1.27e53T + 1.36e106T^{2} \)
23 \( 1 - 2.00e56T + 1.05e113T^{2} \)
29 \( 1 + 5.73e60T + 2.39e121T^{2} \)
31 \( 1 + 1.28e62T + 6.06e123T^{2} \)
37 \( 1 + 1.04e65T + 1.44e130T^{2} \)
41 \( 1 + 9.99e66T + 7.26e133T^{2} \)
43 \( 1 + 9.84e67T + 3.78e135T^{2} \)
47 \( 1 - 1.51e69T + 6.08e138T^{2} \)
53 \( 1 - 6.81e70T + 1.30e143T^{2} \)
59 \( 1 + 1.83e73T + 9.56e146T^{2} \)
61 \( 1 + 2.38e74T + 1.52e148T^{2} \)
67 \( 1 + 5.35e75T + 3.66e151T^{2} \)
71 \( 1 - 3.92e76T + 4.51e153T^{2} \)
73 \( 1 - 2.57e77T + 4.52e154T^{2} \)
79 \( 1 - 1.96e78T + 3.18e157T^{2} \)
83 \( 1 - 4.85e79T + 1.92e159T^{2} \)
89 \( 1 + 1.41e81T + 6.30e161T^{2} \)
97 \( 1 - 4.22e82T + 7.98e164T^{2} \)
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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

−12.56514935772588569474495472788, −11.80179368183374172716918703218, −10.51144590641007809546441433925, −9.446593554235572662888117061235, −7.39198553362586102701435136574, −6.42883167997121748803007677234, −5.24326065145192439281204632879, −3.59039794397489261291743093431, −1.60325493786589936209995264031, −0.11179335629051929744315893201, 0.11179335629051929744315893201, 1.60325493786589936209995264031, 3.59039794397489261291743093431, 5.24326065145192439281204632879, 6.42883167997121748803007677234, 7.39198553362586102701435136574, 9.446593554235572662888117061235, 10.51144590641007809546441433925, 11.80179368183374172716918703218, 12.56514935772588569474495472788

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