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Properties

Label 2-468-117.110-c1-0-8
Degree $2$
Conductor $468$
Sign $0.999 + 0.0331i$
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  + (1.58 − 0.704i)3-s + (−0.321 + 1.20i)5-s + (1.14 + 1.14i)7-s + (2.00 − 2.23i)9-s + (1.64 + 0.441i)11-s + (0.484 − 3.57i)13-s + (0.337 + 2.12i)15-s + (3.21 + 5.56i)17-s + (−6.42 − 1.72i)19-s + (2.62 + 1.00i)21-s + 6.83·23-s + (2.99 + 1.72i)25-s + (1.60 − 4.94i)27-s + (−4.45 + 2.57i)29-s + (3.17 + 0.850i)31-s + ⋯
L(s)  = 1  + (0.913 − 0.407i)3-s + (−0.143 + 0.536i)5-s + (0.433 + 0.433i)7-s + (0.668 − 0.743i)9-s + (0.497 + 0.133i)11-s + (0.134 − 0.990i)13-s + (0.0871 + 0.548i)15-s + (0.779 + 1.34i)17-s + (−1.47 − 0.394i)19-s + (0.572 + 0.219i)21-s + 1.42·23-s + (0.598 + 0.345i)25-s + (0.308 − 0.951i)27-s + (−0.828 + 0.478i)29-s + (0.570 + 0.152i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.999 + 0.0331i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.999 + 0.0331i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99940 - 0.0331811i\)
\(L(\frac12)\) \(\approx\) \(1.99940 - 0.0331811i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 0.704i)T \)
13 \( 1 + (-0.484 + 3.57i)T \)
good5 \( 1 + (0.321 - 1.20i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.14 - 1.14i)T + 7iT^{2} \)
11 \( 1 + (-1.64 - 0.441i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.21 - 5.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.42 + 1.72i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.83T + 23T^{2} \)
29 \( 1 + (4.45 - 2.57i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.17 - 0.850i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.96 - 1.33i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.92 + 8.92i)T + 41iT^{2} \)
43 \( 1 + 8.50iT - 43T^{2} \)
47 \( 1 + (-0.158 - 0.591i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 6.07iT - 53T^{2} \)
59 \( 1 + (-0.757 - 2.82i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 6.58T + 61T^{2} \)
67 \( 1 + (-0.201 + 0.201i)T - 67iT^{2} \)
71 \( 1 + (2.86 - 10.6i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.3 + 10.3i)T + 73iT^{2} \)
79 \( 1 + (7.05 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.95 - 0.522i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.07 - 4.01i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.59 + 3.59i)T - 97iT^{2} \)
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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

−10.82232005390366756935416282333, −10.27431077476134260931825305548, −8.813842894295338637942536618309, −8.543050924908584876353478481218, −7.37516940836908928081740616398, −6.62846804780018057584330306319, −5.38801394093913558440332852655, −3.89439237156650396991003263818, −2.95114562466341682857498281848, −1.62056449157431761427609639350, 1.51869421713071493016977119904, 3.06627407364741113271574654121, 4.30875318248821280527669825002, 4.90447620283322483938527401539, 6.57416914708029710257186320262, 7.54661624083291162035162274824, 8.495711022501168519168979833233, 9.130562689775487092869443925812, 9.968746128032451267760833190609, 11.01013532552899601807105708185

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