Properties

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

Origins

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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  =  $(2,\ 633,\ (\ :0),\ 0.751 + 0.660i)$
$L(\frac{1}{2})$  $\approx$  $0.7896170618$
$L(\frac12)$  $\approx$  $0.7896170618$
$L(1)$   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 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−10.48327844638390701405541912317, −9.749293794376488307719818789909, −8.840198935546719210961999846036, −8.122740057451222617644694114300, −7.64432463450359418652160279602, −6.23054457102105323372141301629, −5.41904430229142979059166833678, −3.68982596198123419253566341173, −2.90008629134652037831146540343, −1.33545731319641007077210955580, 1.55753865669831610511655494103, 3.68582118115226648217656685882, 4.25708827139220254382524781558, 4.89769086620307088203601061944, 7.00711151655427867585610853676, 7.63080925363054826637619306619, 8.307436141523274116357340759762, 9.236120068661618813362696383699, 9.707256983215058832184288000992, 10.47876716676265304770496524190

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