Properties

Degree 2
Conductor $ 3 \cdot 211 $
Sign $0.446 - 0.895i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 1.30i)2-s + (−0.309 − 0.951i)3-s + (−0.500 + 1.53i)4-s + (0.951 + 0.309i)5-s + (0.951 − 1.30i)6-s + (−0.809 − 0.587i)7-s + (−0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s + (0.499 + 1.53i)10-s + (0.587 + 0.190i)11-s + 1.61·12-s + (0.809 + 0.587i)13-s − 1.61i·14-s − 0.999i·15-s + (−0.363 + 0.5i)17-s + (−1.53 − 0.499i)18-s + ⋯
L(s)  = 1  + (0.951 + 1.30i)2-s + (−0.309 − 0.951i)3-s + (−0.500 + 1.53i)4-s + (0.951 + 0.309i)5-s + (0.951 − 1.30i)6-s + (−0.809 − 0.587i)7-s + (−0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s + (0.499 + 1.53i)10-s + (0.587 + 0.190i)11-s + 1.61·12-s + (0.809 + 0.587i)13-s − 1.61i·14-s − 0.999i·15-s + (−0.363 + 0.5i)17-s + (−1.53 − 0.499i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(633\)    =    \(3 \cdot 211\)
\( \varepsilon \)  =  $0.446 - 0.895i$
motivic weight  =  \(0\)
character  :  $\chi_{633} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 633,\ (\ :0),\ 0.446 - 0.895i)$
$L(\frac{1}{2})$  $\approx$  $1.421956617$
$L(\frac12)$  $\approx$  $1.421956617$
$L(1)$   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 \( 1 + (0.309 + 0.951i)T \)
211 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−11.04920192789409707341862641791, −10.10256627431058530383188442451, −8.890836905227442845437328830976, −7.972431015819357613275415493929, −6.85357702034016056863822330750, −6.35959167860760197244198906526, −6.11236780284187645335449138947, −4.73132513355169434220410220191, −3.64388995303213578945049454162, −2.00782974788260612474060929786, 1.77678632430681360936367034764, 3.17123121487354813499534491146, 3.78506068125476433788374115224, 5.07175107855881914200910570760, 5.68575131936137003730358563787, 6.45342988845090995133702912632, 8.590991545450273160908410563832, 9.358753712442343544619079971037, 10.01154221416378726854235507356, 10.68639964805696441621612535755

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