Properties

Degree 2
Conductor $ 3 \cdot 211 $
Sign $0.751 + 0.660i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 + 0.190i)2-s + (0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (0.587 − 0.809i)5-s + (0.587 − 0.190i)6-s + (0.309 + 0.951i)7-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (0.5 − 0.363i)10-s + (−0.951 + 1.30i)11-s − 0.618·12-s + (−0.309 − 0.951i)13-s + 0.618i·14-s i·15-s + (−1.53 + 0.5i)17-s + (0.363 − 0.5i)18-s + ⋯
L(s)  = 1  + (0.587 + 0.190i)2-s + (0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (0.587 − 0.809i)5-s + (0.587 − 0.190i)6-s + (0.309 + 0.951i)7-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (0.5 − 0.363i)10-s + (−0.951 + 1.30i)11-s − 0.618·12-s + (−0.309 − 0.951i)13-s + 0.618i·14-s i·15-s + (−1.53 + 0.5i)17-s + (0.363 − 0.5i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(633\)    =    \(3 \cdot 211\)
\( \varepsilon \)  =  $0.751 + 0.660i$
motivic weight  =  \(0\)
character  :  $\chi_{633} (188, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 633,\ (\ :0),\ 0.751 + 0.660i)$
$L(\frac{1}{2})$  $\approx$  $1.424903184$
$L(\frac12)$  $\approx$  $1.424903184$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;211\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{3,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + (-0.809 + 0.587i)T \)
211 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
11 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.42146647802318201931377819982, −9.623608358695255102892517647332, −8.920286659729656062732035206484, −8.241507329420535546343917990844, −7.14651444386403013070833566956, −5.95876125910742687971610051907, −5.19983729813939363211756870668, −4.39724917669075648191804250101, −2.81098382548455277816684619370, −1.68981350742257403663191874460, 2.57607739656722820028388863709, 3.10575332720359491674074358631, 4.44885057391912521562496132714, 4.90847086402005982810457730465, 6.43839664091665218565118665229, 7.42700049860312932566113912544, 8.473595037881118737030248496591, 9.093789861059388216814220409667, 10.08687028025800864930051571447, 11.00809876593834946538447205645

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