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Properties

Label 2-468-52.51-c0-0-0
Degree $2$
Conductor $468$
Sign $1$
Analytic cond. $0.233562$
Root an. cond. $0.483282$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + 2i·5-s + i·8-s + 2·10-s + 13-s + 16-s − 2i·20-s − 3·25-s − i·26-s − i·32-s − 2·40-s − 2i·41-s − 49-s + 3i·50-s + ⋯
L(s)  = 1  − i·2-s − 4-s + 2i·5-s + i·8-s + 2·10-s + 13-s + 16-s − 2i·20-s − 3·25-s − i·26-s − i·32-s − 2·40-s − 2i·41-s − 49-s + 3i·50-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.233562\)
Root analytic conductor: \(0.483282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7889822930\)
\(L(\frac12)\) \(\approx\) \(0.7889822930\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
13 \( 1 - T \)
good5 \( 1 - 2iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 2iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.16416566599182160015947254535, −10.50989477631144851213797682503, −9.871993969996970715085148754548, −8.718394682031698263175562633973, −7.63209452562785546442834820896, −6.61027724732671725095969472479, −5.61929664722591132016659778564, −3.95166053871250624767144015994, −3.20641668342072801590630720183, −2.10827621551922016124207030190, 1.21066888972617230624490292401, 3.84361350604592800465599760222, 4.75549649486906114033186671544, 5.53811160124869520670393195497, 6.47856971826253364322563561530, 7.925384505200030158799069736783, 8.407532652648921760864359086178, 9.195643625963417459633044764274, 9.901938763572835733248004799793, 11.40966930484865659583781796671

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