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Properties

Label 2-468-117.94-c1-0-1
Degree $2$
Conductor $468$
Sign $-0.987 - 0.155i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (0.180 + 1.72i)3-s + (0.0380 − 0.0658i)5-s − 3.72·7-s + (−2.93 + 0.620i)9-s + (0.561 − 0.972i)11-s + (−2.60 + 2.49i)13-s + (0.120 + 0.0536i)15-s + (−4.03 + 6.99i)17-s + (1.22 − 2.12i)19-s + (−0.670 − 6.41i)21-s − 6.81·23-s + (2.49 + 4.32i)25-s + (−1.59 − 4.94i)27-s + (1.52 − 2.64i)29-s + (4.54 − 7.86i)31-s + ⋯
L(s)  = 1  + (0.103 + 0.994i)3-s + (0.0170 − 0.0294i)5-s − 1.40·7-s + (−0.978 + 0.206i)9-s + (0.169 − 0.293i)11-s + (−0.722 + 0.691i)13-s + (0.0310 + 0.0138i)15-s + (−0.978 + 1.69i)17-s + (0.281 − 0.487i)19-s + (−0.146 − 1.40i)21-s − 1.42·23-s + (0.499 + 0.865i)25-s + (−0.307 − 0.951i)27-s + (0.283 − 0.490i)29-s + (0.815 − 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-0.987 - 0.155i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ -0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0464105 + 0.592199i\)
\(L(\frac12)\) \(\approx\) \(0.0464105 + 0.592199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.180 - 1.72i)T \)
13 \( 1 + (2.60 - 2.49i)T \)
good5 \( 1 + (-0.0380 + 0.0658i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.72T + 7T^{2} \)
11 \( 1 + (-0.561 + 0.972i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.03 - 6.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.22 + 2.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.81T + 23T^{2} \)
29 \( 1 + (-1.52 + 2.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.54 + 7.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.10 - 7.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 6.33T + 43T^{2} \)
47 \( 1 + (-1.74 - 3.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.55T + 53T^{2} \)
59 \( 1 + (-6.54 - 11.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.28T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + (-1.58 + 2.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.39T + 73T^{2} \)
79 \( 1 + (-5.40 - 9.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.01 - 5.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.31 - 10.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−11.35403753014243765529286202912, −10.33913709256668017834649039129, −9.705262032090401265656517748553, −9.023990622642246046818020539855, −8.031836521261380168007982582816, −6.56033731010261636243383244122, −5.96503710730599944547396287662, −4.51614702763014564281116320946, −3.71353453952593091213503857827, −2.51056265409611750285476680158, 0.33294979670938147981893926902, 2.37624173872002100188532951167, 3.29147428284938417817348088839, 4.95775186942399333626307144152, 6.22006752714344719846051922359, 6.85781022720057925867886336471, 7.68800695698570038667823787968, 8.818062231489884903103341473749, 9.671456187475595054711274136653, 10.49898814525698666750655659364

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