Properties

Label 2-480-8.5-c1-0-1
Degree $2$
Conductor $480$
Sign $-0.707 - 0.707i$
Analytic cond. $3.83281$
Root an. cond. $1.95775$
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  + i·3-s + i·5-s − 2·7-s − 9-s + 4i·11-s − 15-s − 6·17-s + 4i·19-s − 2i·21-s + 4·23-s − 25-s i·27-s + 6i·29-s − 10·31-s − 4·33-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.447i·5-s − 0.755·7-s − 0.333·9-s + 1.20i·11-s − 0.258·15-s − 1.45·17-s + 0.917i·19-s − 0.436i·21-s + 0.834·23-s − 0.200·25-s − 0.192i·27-s + 1.11i·29-s − 1.79·31-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350034 + 0.845057i\)
\(L(\frac12)\) \(\approx\) \(0.350034 + 0.845057i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 - iT \)
good7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.96918157989956272974929345013, −10.57768525034885168649416590341, −9.433301718320780963657626634170, −9.023131166919874875061730635826, −7.51388341318594087755656447984, −6.79602086509534910021685230393, −5.70573512328631439719657914075, −4.50758744703912256829584128254, −3.52668710357401300722916867974, −2.21615428018457077783170334966, 0.53302017291401782773935868116, 2.39246308622132532311376050280, 3.63797042099591533503057469427, 5.01336486217339172908995425412, 6.14842491138407039726947953797, 6.83626671141431444491466164445, 7.993488914127447398452585955021, 8.923889297823320942192708446808, 9.490365955720418167979807092638, 11.01760013051485546990871307691

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