Properties

Label 2-462-231.95-c1-0-9
Degree $2$
Conductor $462$
Sign $-0.951 + 0.308i$
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  + (−0.104 + 0.994i)2-s + (0.110 + 1.72i)3-s + (−0.978 − 0.207i)4-s + (−1.75 + 3.94i)5-s + (−1.73 − 0.0705i)6-s + (2.64 + 0.0650i)7-s + (0.309 − 0.951i)8-s + (−2.97 + 0.382i)9-s + (−3.73 − 2.15i)10-s + (3.29 + 0.387i)11-s + (0.251 − 1.71i)12-s + (−2.24 + 3.09i)13-s + (−0.341 + 2.62i)14-s + (−7.01 − 2.59i)15-s + (0.913 + 0.406i)16-s + (−0.195 − 1.86i)17-s + ⋯
L(s)  = 1  + (−0.0739 + 0.703i)2-s + (0.0639 + 0.997i)3-s + (−0.489 − 0.103i)4-s + (−0.785 + 1.76i)5-s + (−0.706 − 0.0288i)6-s + (0.999 + 0.0245i)7-s + (0.109 − 0.336i)8-s + (−0.991 + 0.127i)9-s + (−1.18 − 0.682i)10-s + (0.993 + 0.116i)11-s + (0.0724 − 0.494i)12-s + (−0.623 + 0.858i)13-s + (−0.0911 + 0.701i)14-s + (−1.81 − 0.671i)15-s + (0.228 + 0.101i)16-s + (−0.0474 − 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.951 + 0.308i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178625 - 1.13169i\)
\(L(\frac12)\) \(\approx\) \(0.178625 - 1.13169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 - 0.994i)T \)
3 \( 1 + (-0.110 - 1.72i)T \)
7 \( 1 + (-2.64 - 0.0650i)T \)
11 \( 1 + (-3.29 - 0.387i)T \)
good5 \( 1 + (1.75 - 3.94i)T + (-3.34 - 3.71i)T^{2} \)
13 \( 1 + (2.24 - 3.09i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.195 + 1.86i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (-0.385 - 1.81i)T + (-17.3 + 7.72i)T^{2} \)
23 \( 1 + (-5.76 + 3.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.60 + 4.92i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.16 + 0.519i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-3.11 + 3.45i)T + (-3.86 - 36.7i)T^{2} \)
41 \( 1 + (2.90 - 8.94i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.90iT - 43T^{2} \)
47 \( 1 + (1.28 + 6.05i)T + (-42.9 + 19.1i)T^{2} \)
53 \( 1 + (-5.16 - 11.5i)T + (-35.4 + 39.3i)T^{2} \)
59 \( 1 + (-0.0588 + 0.276i)T + (-53.8 - 23.9i)T^{2} \)
61 \( 1 + (3.90 - 8.77i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (3.82 - 6.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.878 - 1.20i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.25 + 5.92i)T + (-66.6 - 29.6i)T^{2} \)
79 \( 1 + (-4.12 - 0.433i)T + (77.2 + 16.4i)T^{2} \)
83 \( 1 + (4.25 - 3.08i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.326 + 0.188i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.32 + 2.41i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.47481560599496197123233960815, −10.64063406886673713413258378589, −9.764693122860352602022261579975, −8.815548462273600158488816200335, −7.76751339765527867447204108187, −7.01850609249468000299971445891, −6.09112831678068214360689327621, −4.63947887480686838367367987350, −3.97873583083535414264211828553, −2.67664123645642413845522657614, 0.78708630729016095726032503755, 1.69245835232227612409657887569, 3.51153923162082092697294535942, 4.78801077539104363428364317572, 5.44887605962724352298950121926, 7.20236873234385723633188634633, 8.063891266211788252879575790438, 8.681361148540975120198214026435, 9.340329424751673499326828668001, 10.96943010026056941604351203460

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