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Properties

Label 2-480-24.11-c1-0-11
Degree $2$
Conductor $480$
Sign $0.872 + 0.489i$
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  + (1.71 − 0.218i)3-s + 5-s − 3.64i·7-s + (2.90 − 0.750i)9-s + 5.07i·11-s − 1.70i·13-s + (1.71 − 0.218i)15-s − 4.08i·17-s + 1.26·19-s + (−0.796 − 6.26i)21-s − 4.70·23-s + 25-s + (4.82 − 1.92i)27-s + 1.06·29-s + 4.86i·31-s + ⋯
L(s)  = 1  + (0.992 − 0.126i)3-s + 0.447·5-s − 1.37i·7-s + (0.968 − 0.250i)9-s + 1.52i·11-s − 0.473i·13-s + (0.443 − 0.0564i)15-s − 0.989i·17-s + 0.290·19-s + (−0.173 − 1.36i)21-s − 0.980·23-s + 0.200·25-s + (0.928 − 0.370i)27-s + 0.197·29-s + 0.874i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.872 + 0.489i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 0.872 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04249 - 0.533845i\)
\(L(\frac12)\) \(\approx\) \(2.04249 - 0.533845i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.218i)T \)
5 \( 1 - T \)
good7 \( 1 + 3.64iT - 7T^{2} \)
11 \( 1 - 5.07iT - 11T^{2} \)
13 \( 1 + 1.70iT - 13T^{2} \)
17 \( 1 + 4.08iT - 17T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 - 1.06T + 29T^{2} \)
31 \( 1 - 4.86iT - 31T^{2} \)
37 \( 1 - 7.56iT - 37T^{2} \)
41 \( 1 - 1.50iT - 41T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 8.87T + 53T^{2} \)
59 \( 1 + 0.788iT - 59T^{2} \)
61 \( 1 + 0.627iT - 61T^{2} \)
67 \( 1 + 4.18T + 67T^{2} \)
71 \( 1 + 6.21T + 71T^{2} \)
73 \( 1 + 4.21T + 73T^{2} \)
79 \( 1 + 0.992iT - 79T^{2} \)
83 \( 1 - 7.72iT - 83T^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 + 7.40T + 97T^{2} \)
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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

−10.51463137612131508799968299804, −9.989221997897212500469617104819, −9.319273916181930613768334094006, −8.045462822982085481872983532818, −7.32931977002084959639100829874, −6.66859460409016473504628609774, −4.93795485293028090762054253710, −4.06313436074399490046305141234, −2.80265078393113267883044650614, −1.43354141810858636021574041974, 1.90199984137495785113130321800, 2.93221897186327431717375345159, 4.07118420215339861950875112193, 5.60345162431722067458840274974, 6.21981201889557157432271763674, 7.70337577639865026138749595408, 8.646116403123228488759414707233, 9.006709577209535693668191047513, 9.996491547975187987607031676488, 10.99783949195867194706530651123

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