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Properties

Label 2-462-33.8-c1-0-19
Degree $2$
Conductor $462$
Sign $0.758 + 0.651i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (1.39 + 1.03i)3-s + (0.309 − 0.951i)4-s + (1.54 − 2.11i)5-s + (1.73 + 0.0158i)6-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s + (0.874 + 2.86i)9-s − 2.62i·10-s + (−0.567 − 3.26i)11-s + (1.41 − 1.00i)12-s + (−0.538 − 0.741i)13-s + (−0.951 + 0.309i)14-s + (4.32 − 1.36i)15-s + (−0.809 − 0.587i)16-s + (3.44 + 2.50i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.803 + 0.595i)3-s + (0.154 − 0.475i)4-s + (0.688 − 0.947i)5-s + (0.707 + 0.00648i)6-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s + (0.291 + 0.956i)9-s − 0.828i·10-s + (−0.171 − 0.985i)11-s + (0.407 − 0.290i)12-s + (−0.149 − 0.205i)13-s + (−0.254 + 0.0825i)14-s + (1.11 − 0.351i)15-s + (−0.202 − 0.146i)16-s + (0.835 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.758 + 0.651i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42668 - 0.899669i\)
\(L(\frac12)\) \(\approx\) \(2.42668 - 0.899669i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-1.39 - 1.03i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (0.567 + 3.26i)T \)
good5 \( 1 + (-1.54 + 2.11i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (0.538 + 0.741i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.44 - 2.50i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.49 - 0.485i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.00iT - 23T^{2} \)
29 \( 1 + (2.38 - 7.35i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.41 - 1.02i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.04 - 6.30i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.42 + 4.39i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.8iT - 43T^{2} \)
47 \( 1 + (-5.43 + 1.76i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.38 - 8.79i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.219 + 0.0714i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.88 - 10.8i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.86T + 67T^{2} \)
71 \( 1 + (-4.47 + 6.15i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (13.6 + 4.42i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.66 + 2.28i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.4 + 8.33i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.91iT - 89T^{2} \)
97 \( 1 + (13.7 - 10.0i)T + (29.9 - 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61778867364542193770301061160, −10.23854885491579124430611261405, −9.073274801020876319615003791746, −8.655600605839143185008211323345, −7.35346625216665164153469441276, −5.77608133649289660498544908661, −5.20561287747709028755525367929, −3.92900061659771870358287242996, −3.03200612713323220860975057810, −1.55500520314995309892951819198, 2.16274231764255835431321169804, 2.95391396769922003331076129201, 4.24466436864272701005943851940, 5.73614590031897154691377627831, 6.64974545995287773054552693170, 7.26942923825772094152845161448, 8.163750035444143170753902100591, 9.500205462241381877326150806313, 9.958924487905723842767071962899, 11.30096126556282812802223704894

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