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 − 2.10e19·3-s + 1.20e24·4-s − 2.29e27·5-s + 2.31e31·6-s + 2.01e34·7-s − 1.32e36·8-s − 1.36e36·9-s + 2.52e39·10-s − 1.36e40·11-s − 2.54e43·12-s − 1.42e45·13-s − 2.21e46·14-s + 4.83e46·15-s + 1.46e48·16-s − 2.49e49·17-s + 1.49e48·18-s + 3.58e51·19-s − 2.77e51·20-s − 4.24e53·21-s + 1.49e52·22-s − 1.21e54·23-s + 2.79e55·24-s − 4.08e56·25-s + 1.56e57·26-s + 9.35e57·27-s + 2.44e58·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.998·3-s + 0.5·4-s − 0.112·5-s + 0.706·6-s + 1.19·7-s − 0.353·8-s − 0.00307·9-s + 0.0798·10-s − 0.00907·11-s − 0.499·12-s − 1.09·13-s − 0.847·14-s + 0.112·15-s + 0.250·16-s − 0.365·17-s + 0.00217·18-s + 0.581·19-s − 0.0564·20-s − 1.19·21-s + 0.00641·22-s − 0.0859·23-s + 0.353·24-s − 0.987·25-s + 0.773·26-s + 1.00·27-s + 0.599·28-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 \( 1 + 2.10e19T + 4.43e38T^{2} \)
5 \( 1 + 2.29e27T + 4.13e56T^{2} \)
7 \( 1 - 2.01e34T + 2.83e68T^{2} \)
11 \( 1 + 1.36e40T + 2.25e84T^{2} \)
13 \( 1 + 1.42e45T + 1.69e90T^{2} \)
17 \( 1 + 2.49e49T + 4.63e99T^{2} \)
19 \( 1 - 3.58e51T + 3.79e103T^{2} \)
23 \( 1 + 1.21e54T + 1.99e110T^{2} \)
29 \( 1 - 7.48e58T + 2.84e118T^{2} \)
31 \( 1 - 4.18e60T + 6.31e120T^{2} \)
37 \( 1 + 4.16e63T + 1.05e127T^{2} \)
41 \( 1 - 2.46e65T + 4.32e130T^{2} \)
43 \( 1 - 4.35e65T + 2.04e132T^{2} \)
47 \( 1 - 5.69e67T + 2.75e135T^{2} \)
53 \( 1 + 4.75e69T + 4.63e139T^{2} \)
59 \( 1 - 8.16e71T + 2.74e143T^{2} \)
61 \( 1 - 2.39e72T + 4.08e144T^{2} \)
67 \( 1 + 4.34e72T + 8.16e147T^{2} \)
71 \( 1 + 1.27e75T + 8.95e149T^{2} \)
73 \( 1 + 2.61e75T + 8.49e150T^{2} \)
79 \( 1 + 5.46e76T + 5.10e153T^{2} \)
83 \( 1 + 3.44e77T + 2.78e155T^{2} \)
89 \( 1 + 1.44e79T + 7.95e157T^{2} \)
97 \( 1 + 3.90e80T + 8.48e160T^{2} \)
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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

−12.06536689271228537383935783759, −11.33276885695560761944184492947, −10.08397471696831798285837330471, −8.421978727899651524210412131409, −7.20307415043015495547717835981, −5.70631700038199326996320614006, −4.58984046791382573935465836002, −2.47516467190448767621510586119, −1.11177672784169054483642633318, 0, 1.11177672784169054483642633318, 2.47516467190448767621510586119, 4.58984046791382573935465836002, 5.70631700038199326996320614006, 7.20307415043015495547717835981, 8.421978727899651524210412131409, 10.08397471696831798285837330471, 11.33276885695560761944184492947, 12.06536689271228537383935783759

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