Properties

Degree 2
Conductor $ 2^{7} \cdot 19 $
Sign $i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·3-s − 5-s + 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s − 1.41i·33-s − 35-s + 1.41i·37-s − 1.41i·41-s + ⋯
L(s)  = 1  − 1.41i·3-s − 5-s + 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s − 1.41i·33-s − 35-s + 1.41i·37-s − 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2432\)    =    \(2^{7} \cdot 19\)
\( \varepsilon \)  =  $i$
motivic weight  =  \(0\)
character  :  $\chi_{2432} (1025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2432,\ (\ :0),\ i)$
$L(\frac{1}{2})$  $\approx$  $1.227024270$
$L(\frac12)$  $\approx$  $1.227024270$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.491825895088522119862322109607, −8.101600362273263136931656871480, −7.41026071020686152316376256933, −6.92236239437042172429745359214, −5.97652372593103427789680967519, −5.04787558198727184179532020758, −4.05007910976000626973555023779, −3.13396020373967825159368495513, −1.79225205749975178118609305358, −1.02223138353239144806784506481, 1.39103162093925498407498392169, 3.12035050977351861637054268404, 3.81956715260758046083100426242, 4.44063626358879832058191476387, 5.14919372078583084671490445168, 5.98367756253079699687046587307, 7.34030527649772624873804638411, 7.78381254635765515907635843900, 8.705667051627909153195451006235, 9.344948254784760498386269547587

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