Properties

Degree 2
Conductor 2
Sign $-1$
Motivic weight 81
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.09e12·2-s + 2.69e19·3-s + 1.20e24·4-s + 1.65e28·5-s − 2.96e31·6-s + 2.15e33·7-s − 1.32e36·8-s + 2.82e38·9-s − 1.82e40·10-s − 3.85e40·11-s + 3.25e43·12-s − 8.46e43·13-s − 2.36e45·14-s + 4.46e47·15-s + 1.46e48·16-s − 7.68e49·17-s − 3.10e50·18-s − 7.37e51·19-s + 2.00e52·20-s + 5.80e52·21-s + 4.23e52·22-s − 1.05e55·23-s − 3.58e55·24-s − 1.39e56·25-s + 9.31e55·26-s − 4.34e57·27-s + 2.60e57·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.27·3-s + 0.5·4-s + 0.814·5-s − 0.904·6-s + 0.127·7-s − 0.353·8-s + 0.636·9-s − 0.575·10-s − 0.0256·11-s + 0.639·12-s − 0.0650·13-s − 0.0904·14-s + 1.04·15-s + 0.250·16-s − 1.12·17-s − 0.450·18-s − 1.19·19-s + 0.407·20-s + 0.163·21-s + 0.0181·22-s − 0.745·23-s − 0.452·24-s − 0.337·25-s + 0.0459·26-s − 0.465·27-s + 0.0639·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(82-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+81/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(81\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2,\ (\ :81/2),\ -1)$
$L(41)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{83}{2})$   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 \( 1 + 1.09e12T \)
good3 \( 1 - 2.69e19T + 4.43e38T^{2} \)
5 \( 1 - 1.65e28T + 4.13e56T^{2} \)
7 \( 1 - 2.15e33T + 2.83e68T^{2} \)
11 \( 1 + 3.85e40T + 2.25e84T^{2} \)
13 \( 1 + 8.46e43T + 1.69e90T^{2} \)
17 \( 1 + 7.68e49T + 4.63e99T^{2} \)
19 \( 1 + 7.37e51T + 3.79e103T^{2} \)
23 \( 1 + 1.05e55T + 1.99e110T^{2} \)
29 \( 1 + 1.79e59T + 2.84e118T^{2} \)
31 \( 1 + 4.86e60T + 6.31e120T^{2} \)
37 \( 1 - 2.30e63T + 1.05e127T^{2} \)
41 \( 1 + 1.35e65T + 4.32e130T^{2} \)
43 \( 1 - 2.10e66T + 2.04e132T^{2} \)
47 \( 1 - 1.03e68T + 2.75e135T^{2} \)
53 \( 1 - 2.99e69T + 4.63e139T^{2} \)
59 \( 1 + 5.68e71T + 2.74e143T^{2} \)
61 \( 1 - 5.44e71T + 4.08e144T^{2} \)
67 \( 1 + 5.75e73T + 8.16e147T^{2} \)
71 \( 1 - 4.64e74T + 8.95e149T^{2} \)
73 \( 1 - 4.70e75T + 8.49e150T^{2} \)
79 \( 1 + 4.20e76T + 5.10e153T^{2} \)
83 \( 1 + 7.46e77T + 2.78e155T^{2} \)
89 \( 1 - 5.59e78T + 7.95e157T^{2} \)
97 \( 1 + 1.28e80T + 8.48e160T^{2} \)
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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

−12.96446727965251858602492250770, −10.91540996963695945676992130567, −9.494518375178867106013539250488, −8.723154818811695291405901533350, −7.47186979821420211745410067144, −5.94217770458994949981878820727, −3.93042105477405562264675147715, −2.36424398822134285468909051906, −1.85240261465840691084710946396, 0, 1.85240261465840691084710946396, 2.36424398822134285468909051906, 3.93042105477405562264675147715, 5.94217770458994949981878820727, 7.47186979821420211745410067144, 8.723154818811695291405901533350, 9.494518375178867106013539250488, 10.91540996963695945676992130567, 12.96446727965251858602492250770

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