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Properties

Label 2-462-33.8-c1-0-22
Degree $2$
Conductor $462$
Sign $-0.366 + 0.930i$
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 + (0.677 − 1.59i)3-s + (0.309 − 0.951i)4-s + (−0.456 + 0.628i)5-s + (−0.389 − 1.68i)6-s + (0.951 + 0.309i)7-s + (−0.309 − 0.951i)8-s + (−2.08 − 2.15i)9-s + 0.776i·10-s + (−0.889 − 3.19i)11-s + (−1.30 − 1.13i)12-s + (−2.43 − 3.34i)13-s + (0.951 − 0.309i)14-s + (0.692 + 1.15i)15-s + (−0.809 − 0.587i)16-s + (1.26 + 0.918i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.391 − 0.920i)3-s + (0.154 − 0.475i)4-s + (−0.204 + 0.281i)5-s + (−0.158 − 0.689i)6-s + (0.359 + 0.116i)7-s + (−0.109 − 0.336i)8-s + (−0.694 − 0.719i)9-s + 0.245i·10-s + (−0.268 − 0.963i)11-s + (−0.377 − 0.328i)12-s + (−0.674 − 0.928i)13-s + (0.254 − 0.0825i)14-s + (0.178 + 0.297i)15-s + (−0.202 − 0.146i)16-s + (0.306 + 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.366 + 0.930i$
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: \((2,\ 462,\ (\ :1/2),\ -0.366 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14456 - 1.68027i\)
\(L(\frac12)\) \(\approx\) \(1.14456 - 1.68027i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.677 + 1.59i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (0.889 + 3.19i)T \)
good5 \( 1 + (0.456 - 0.628i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.43 + 3.34i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.26 - 0.918i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-7.38 + 2.39i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.66iT - 23T^{2} \)
29 \( 1 + (2.02 - 6.21i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.15 + 3.02i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.50 + 7.70i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.798 + 2.45i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 + (12.2 - 3.97i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.19 - 9.90i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-8.69 - 2.82i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.0606 - 0.0834i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + (3.77 - 5.19i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (7.59 + 2.46i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.807 - 1.11i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.532 - 0.386i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.0iT - 89T^{2} \)
97 \( 1 + (-10.3 + 7.49i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.23255532210758936474418128282, −9.948987458490685973360943623229, −8.940299395285074565612123883602, −7.79772205301165890564721833633, −7.25508853793555828477318891041, −5.89847310371178199536939939115, −5.18446441885889581641666635285, −3.40722783008186498879673993957, −2.76344900356214130626172820212, −1.09777539134892700587652047776, 2.33768496786522905482288012585, 3.69781126453970488482754024049, 4.71848187138875712116385084434, 5.20527784960370721232965242124, 6.73084389364317176197696905038, 7.75349463551455779265510767305, 8.480022631019167806478752800559, 9.719748755365457162518591306484, 10.13291379383051597062045142700, 11.70146887390455211823243197393

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