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Properties

Label 2-468-117.61-c1-0-6
Degree $2$
Conductor $468$
Sign $0.995 + 0.0988i$
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  + (1.20 − 1.24i)3-s + (1.66 + 2.88i)5-s − 0.0102·7-s + (−0.0807 − 2.99i)9-s + (−0.798 − 1.38i)11-s + (3.42 − 1.12i)13-s + (5.60 + 1.42i)15-s + (2.62 + 4.55i)17-s + (0.538 + 0.932i)19-s + (−0.0123 + 0.0126i)21-s − 2.42·23-s + (−3.06 + 5.31i)25-s + (−3.81 − 3.52i)27-s + (2.23 + 3.86i)29-s + (−1.47 − 2.55i)31-s + ⋯
L(s)  = 1  + (0.697 − 0.716i)3-s + (0.746 + 1.29i)5-s − 0.00386·7-s + (−0.0269 − 0.999i)9-s + (−0.240 − 0.416i)11-s + (0.950 − 0.311i)13-s + (1.44 + 0.366i)15-s + (0.637 + 1.10i)17-s + (0.123 + 0.213i)19-s + (−0.00269 + 0.00277i)21-s − 0.505·23-s + (−0.613 + 1.06i)25-s + (−0.735 − 0.677i)27-s + (0.414 + 0.718i)29-s + (−0.265 − 0.459i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.995 + 0.0988i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.995 + 0.0988i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00659 - 0.0994700i\)
\(L(\frac12)\) \(\approx\) \(2.00659 - 0.0994700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.24i)T \)
13 \( 1 + (-3.42 + 1.12i)T \)
good5 \( 1 + (-1.66 - 2.88i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.0102T + 7T^{2} \)
11 \( 1 + (0.798 + 1.38i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.62 - 4.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.538 - 0.932i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.42T + 23T^{2} \)
29 \( 1 + (-2.23 - 3.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.47 + 2.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.21 + 7.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.41T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (-4.91 + 8.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.27T + 53T^{2} \)
59 \( 1 + (5.53 - 9.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 3.39T + 67T^{2} \)
71 \( 1 + (-1.93 - 3.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + (-0.0422 + 0.0732i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.14 + 12.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.47 - 2.55i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.56T + 97T^{2} \)
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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

−10.81126927512445762104880889563, −10.25214705441537780475860821065, −9.169708526885786509853727730772, −8.188770377333176627227546900684, −7.39695075748291562771544779688, −6.28284360328091059829846949963, −5.86119828226166413910352078528, −3.71735817040255154237619934535, −2.89188172292413898126093683446, −1.65332563335618659572017139500, 1.53804925213446395875339714076, 3.00152195209080689274978266810, 4.44077377424958009402356036815, 5.07419868459816559636348920144, 6.18279539151846858964125615214, 7.74250590922349225311350071624, 8.492087181473465147804943209765, 9.454356933090842652431211018948, 9.718848971037568626924800716920, 10.89429152355795654468669828708

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