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  =  $(2,\ 2432,\ (\ :0),\ -i)$
$L(\frac{1}{2})$  $\approx$  $1.510196287$
$L(\frac12)$  $\approx$  $1.510196287$
$L(1)$   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

−9.250953202723730190323717888269, −8.872935448251216144919976606266, −7.938191712881340308333911530042, −7.09110441015822124429419541493, −5.73688091571638148801460200408, −5.39273852645886723190454549553, −4.67957744370422299727739078929, −3.79306020109889461278852983434, −2.73247974143644809381584538023, −1.68289640088222024848000251785, 1.10537673504947403609078840838, 2.10761664994268906131972631424, 2.59245854390906704341164083930, 4.23152317681079038390802354369, 5.27815568994135145061491950944, 5.94765366757711923672069878061, 6.53998668565952309128283896751, 7.70912239580716861992538208203, 7.87590664909769522954618263760, 8.653094923891008206358254448164

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