Properties

Degree 2
Conductor 2
Sign $-1$
Motivic weight 47
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.38e6·2-s − 1.96e11·3-s + 7.03e13·4-s + 2.06e16·5-s − 1.64e18·6-s − 5.11e19·7-s + 5.90e20·8-s + 1.20e22·9-s + 1.73e23·10-s + 5.29e24·11-s − 1.38e25·12-s − 1.25e26·13-s − 4.29e26·14-s − 4.06e27·15-s + 4.95e27·16-s − 4.48e28·17-s + 1.01e29·18-s − 1.11e30·19-s + 1.45e30·20-s + 1.00e31·21-s + 4.44e31·22-s − 1.78e32·23-s − 1.16e32·24-s − 2.83e32·25-s − 1.04e33·26-s + 2.85e33·27-s − 3.60e33·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.20·3-s + 1/2·4-s + 0.775·5-s − 0.852·6-s − 0.706·7-s + 0.353·8-s + 0.454·9-s + 0.548·10-s + 1.78·11-s − 0.602·12-s − 0.830·13-s − 0.499·14-s − 0.935·15-s + 1/4·16-s − 0.544·17-s + 0.321·18-s − 0.988·19-s + 0.387·20-s + 0.852·21-s + 1.26·22-s − 1.77·23-s − 0.426·24-s − 0.398·25-s − 0.587·26-s + 0.658·27-s − 0.353·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(48-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(47\)
character  :  $\chi_{2} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2,\ (\ :47/2),\ -1)$
$L(24)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{49}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - p^{23} T \)
good3 \( 1 + 809195804 p^{5} T + p^{47} T^{2} \)
5 \( 1 - 165359697351846 p^{3} T + p^{47} T^{2} \)
7 \( 1 + 1044344527283602504 p^{2} T + p^{47} T^{2} \)
11 \( 1 - \)\(43\!\cdots\!12\)\( p^{2} T + p^{47} T^{2} \)
13 \( 1 + \)\(74\!\cdots\!58\)\( p^{2} T + p^{47} T^{2} \)
17 \( 1 + \)\(26\!\cdots\!78\)\( p T + p^{47} T^{2} \)
19 \( 1 + \)\(58\!\cdots\!00\)\( p T + p^{47} T^{2} \)
23 \( 1 + \)\(77\!\cdots\!64\)\( p T + p^{47} T^{2} \)
29 \( 1 + \)\(30\!\cdots\!90\)\( T + p^{47} T^{2} \)
31 \( 1 + \)\(11\!\cdots\!28\)\( T + p^{47} T^{2} \)
37 \( 1 - \)\(16\!\cdots\!42\)\( p T + p^{47} T^{2} \)
41 \( 1 + \)\(86\!\cdots\!58\)\( p T + p^{47} T^{2} \)
43 \( 1 - \)\(78\!\cdots\!76\)\( p T + p^{47} T^{2} \)
47 \( 1 + \)\(22\!\cdots\!36\)\( T + p^{47} T^{2} \)
53 \( 1 + \)\(29\!\cdots\!02\)\( T + p^{47} T^{2} \)
59 \( 1 - \)\(40\!\cdots\!20\)\( T + p^{47} T^{2} \)
61 \( 1 - \)\(57\!\cdots\!42\)\( T + p^{47} T^{2} \)
67 \( 1 + \)\(18\!\cdots\!36\)\( T + p^{47} T^{2} \)
71 \( 1 - \)\(55\!\cdots\!12\)\( T + p^{47} T^{2} \)
73 \( 1 + \)\(79\!\cdots\!02\)\( T + p^{47} T^{2} \)
79 \( 1 - \)\(88\!\cdots\!40\)\( T + p^{47} T^{2} \)
83 \( 1 + \)\(73\!\cdots\!92\)\( T + p^{47} T^{2} \)
89 \( 1 + \)\(82\!\cdots\!90\)\( T + p^{47} T^{2} \)
97 \( 1 + \)\(61\!\cdots\!46\)\( T + p^{47} T^{2} \)
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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

−16.50690140691374186946979193440, −14.41934631887018138093812925902, −12.67467475024587745538068871040, −11.41167712682993798568045806326, −9.670791487927185637443775027015, −6.60017203139532905940607846544, −5.80879569785079717513056325231, −4.09167792003467702657852794553, −1.91234233200451036802198779504, 0, 1.91234233200451036802198779504, 4.09167792003467702657852794553, 5.80879569785079717513056325231, 6.60017203139532905940607846544, 9.670791487927185637443775027015, 11.41167712682993798568045806326, 12.67467475024587745538068871040, 14.41934631887018138093812925902, 16.50690140691374186946979193440

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