Properties

Degree 2
Conductor $ 2 \cdot 117223 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 4

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 4-s − 4·5-s + 3·6-s − 5·7-s − 8-s + 6·9-s + 4·10-s − 6·11-s − 3·12-s − 6·13-s + 5·14-s + 12·15-s + 16-s − 6·17-s − 6·18-s − 8·19-s − 4·20-s + 15·21-s + 6·22-s − 6·23-s + 3·24-s + 11·25-s + 6·26-s − 9·27-s − 5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.78·5-s + 1.22·6-s − 1.88·7-s − 0.353·8-s + 2·9-s + 1.26·10-s − 1.80·11-s − 0.866·12-s − 1.66·13-s + 1.33·14-s + 3.09·15-s + 1/4·16-s − 1.45·17-s − 1.41·18-s − 1.83·19-s − 0.894·20-s + 3.27·21-s + 1.27·22-s − 1.25·23-s + 0.612·24-s + 11/5·25-s + 1.17·26-s − 1.73·27-s − 0.944·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 234446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 234446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(234446\)    =    \(2 \cdot 117223\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{234446} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  4
Selberg data  =  $(2,\ 234446,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;117223\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;117223\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
117223 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.33368034086200, −13.09360406818600, −12.73824472819508, −12.24614205522120, −12.04497230225563, −11.55674326486266, −10.89126786866145, −10.58503012146236, −10.35721937854419, −9.879771989628829, −9.261689397031543, −8.645433260663214, −8.136132709760610, −7.500600983907722, −7.222532857430923, −6.730090190731062, −6.443618701646912, −5.825068468989255, −5.165585013784794, −4.717665714334101, −4.254722279811606, −3.589684506825036, −3.045158921575872, −2.335210734225171, −1.733532247157827, 0, 0, 0, 0, 1.733532247157827, 2.335210734225171, 3.045158921575872, 3.589684506825036, 4.254722279811606, 4.717665714334101, 5.165585013784794, 5.825068468989255, 6.443618701646912, 6.730090190731062, 7.222532857430923, 7.500600983907722, 8.136132709760610, 8.645433260663214, 9.261689397031543, 9.879771989628829, 10.35721937854419, 10.58503012146236, 10.89126786866145, 11.55674326486266, 12.04497230225563, 12.24614205522120, 12.73824472819508, 13.09360406818600, 13.33368034086200

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