Properties

Degree 2
Conductor $ 11 \cdot 233 $
Sign $-1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 1.41i·3-s − 1.00·4-s + 1.41i·5-s + 2.00·6-s − 1.00·9-s − 2.00·10-s − 11-s + 1.41i·12-s + 2.00·15-s − 0.999·16-s − 17-s − 1.41i·18-s + 1.41i·19-s − 1.41i·20-s + ⋯
L(s)  = 1  + 1.41i·2-s − 1.41i·3-s − 1.00·4-s + 1.41i·5-s + 2.00·6-s − 1.00·9-s − 2.00·10-s − 11-s + 1.41i·12-s + 2.00·15-s − 0.999·16-s − 17-s − 1.41i·18-s + 1.41i·19-s − 1.41i·20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2563\)    =    \(11 \cdot 233\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(0\)
character  :  $\chi_{2563} (2562, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2563,\ (\ :0),\ -1)$
$L(\frac{1}{2})$  $\approx$  $0.7085197646$
$L(\frac12)$  $\approx$  $0.7085197646$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{11,\;233\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{11,\;233\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad11 \( 1 + T \)
233 \( 1 - T \)
good2 \( 1 - 1.41iT - T^{2} \)
3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−9.024199886102955549724432401123, −8.198120055933097180651501151626, −7.56992681696179587969796202837, −7.24586468610773280239023326767, −6.50892096593132392100438424575, −6.03133522837937432936503459265, −5.28800620688109138976054973666, −3.91096781329385457098304243135, −2.64741712167278805447573544247, −1.94669311806842822348200736691, 0.42656929302875800099008565060, 2.01167708205642339916872271798, 2.88022517795644054179448284056, 4.02889128610351794862898201543, 4.47807619637250736327595134535, 5.02538466275586459080282852221, 5.97810237262983055390819926905, 7.39169254687569231610171592698, 8.531493529962721411274021462105, 8.966308471757042360398399669638

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