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Properties

Label 2-468-117.94-c1-0-0
Degree $2$
Conductor $468$
Sign $-0.712 - 0.702i$
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  + (−0.674 − 1.59i)3-s + (−0.0842 + 0.145i)5-s − 2.38·7-s + (−2.08 + 2.15i)9-s + (−1.73 + 3.01i)11-s + (−3.24 − 1.57i)13-s + (0.289 + 0.0359i)15-s + (2.47 − 4.28i)17-s + (−3.33 + 5.77i)19-s + (1.61 + 3.80i)21-s − 6.09·23-s + (2.48 + 4.30i)25-s + (4.84 + 1.87i)27-s + (1.83 − 3.18i)29-s + (−1.42 + 2.47i)31-s + ⋯
L(s)  = 1  + (−0.389 − 0.920i)3-s + (−0.0376 + 0.0652i)5-s − 0.902·7-s + (−0.696 + 0.717i)9-s + (−0.524 + 0.908i)11-s + (−0.900 − 0.435i)13-s + (0.0748 + 0.00927i)15-s + (0.600 − 1.04i)17-s + (−0.765 + 1.32i)19-s + (0.351 + 0.830i)21-s − 1.27·23-s + (0.497 + 0.861i)25-s + (0.932 + 0.361i)27-s + (0.341 − 0.590i)29-s + (−0.256 + 0.444i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-0.712 - 0.702i$
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: \((2,\ 468,\ (\ :1/2),\ -0.712 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0328164 + 0.0800379i\)
\(L(\frac12)\) \(\approx\) \(0.0328164 + 0.0800379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.674 + 1.59i)T \)
13 \( 1 + (3.24 + 1.57i)T \)
good5 \( 1 + (0.0842 - 0.145i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.38T + 7T^{2} \)
11 \( 1 + (1.73 - 3.01i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.47 + 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.33 - 5.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.09T + 23T^{2} \)
29 \( 1 + (-1.83 + 3.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.42 - 2.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.36 - 2.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.32T + 41T^{2} \)
43 \( 1 + 6.24T + 43T^{2} \)
47 \( 1 + (4.39 + 7.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 14.0T + 53T^{2} \)
59 \( 1 + (0.411 + 0.712i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + (-2.14 + 3.71i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 + (1.51 + 2.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.944 + 1.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.27 + 3.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.53T + 97T^{2} \)
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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

−11.68480408128798628155151982193, −10.15301298397315779859445502815, −9.994294063912406221275943522939, −8.446799999153046194590634378462, −7.53886535011029926284012940371, −6.87340130027253045338712453749, −5.83662009997965069248002497664, −4.89771513285948844172114066214, −3.22522030388533227383830662694, −1.99565202161489273275970813364, 0.05144507664902152654268454233, 2.72323331229146972143691809622, 3.83121627239776497283419720045, 4.89439217213975385523375374961, 5.96772625742273017593272230271, 6.72261986315563680020863419651, 8.205933064140567450896795160575, 9.004777464312933279683759343800, 10.00973671926206414491099403537, 10.48620247022965844747683800162

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