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Properties

Label 2-480-20.7-c1-0-2
Degree $2$
Conductor $480$
Sign $0.525 - 0.850i$
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  + (−0.707 − 0.707i)3-s + (−2.12 − 0.707i)5-s + (−1 + i)7-s + 1.00i·9-s + 5.41i·11-s + (3.41 − 3.41i)13-s + (0.999 + 2i)15-s + (5.41 + 5.41i)17-s − 4.82·19-s + 1.41·21-s + (4.24 + 4.24i)23-s + (3.99 + 3i)25-s + (0.707 − 0.707i)27-s + 0.242i·29-s − 1.65i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.948 − 0.316i)5-s + (−0.377 + 0.377i)7-s + 0.333i·9-s + 1.63i·11-s + (0.946 − 0.946i)13-s + (0.258 + 0.516i)15-s + (1.31 + 1.31i)17-s − 1.10·19-s + 0.308·21-s + (0.884 + 0.884i)23-s + (0.799 + 0.600i)25-s + (0.136 − 0.136i)27-s + 0.0450i·29-s − 0.297i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749877 + 0.418084i\)
\(L(\frac12)\) \(\approx\) \(0.749877 + 0.418084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (2.12 + 0.707i)T \)
good7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 - 5.41iT - 11T^{2} \)
13 \( 1 + (-3.41 + 3.41i)T - 13iT^{2} \)
17 \( 1 + (-5.41 - 5.41i)T + 17iT^{2} \)
19 \( 1 + 4.82T + 19T^{2} \)
23 \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \)
29 \( 1 - 0.242iT - 29T^{2} \)
31 \( 1 + 1.65iT - 31T^{2} \)
37 \( 1 + (0.585 + 0.585i)T + 37iT^{2} \)
41 \( 1 + 6.48T + 41T^{2} \)
43 \( 1 + (-4.82 - 4.82i)T + 43iT^{2} \)
47 \( 1 + (6.24 - 6.24i)T - 47iT^{2} \)
53 \( 1 + (2.82 - 2.82i)T - 53iT^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 3.17T + 61T^{2} \)
67 \( 1 + (3.65 - 3.65i)T - 67iT^{2} \)
71 \( 1 + 2.34iT - 71T^{2} \)
73 \( 1 + (-7.48 + 7.48i)T - 73iT^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 + (-3.07 - 3.07i)T + 83iT^{2} \)
89 \( 1 - 3.65iT - 89T^{2} \)
97 \( 1 + (-0.656 - 0.656i)T + 97iT^{2} \)
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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

−11.13976385482226363130461554817, −10.40891539312751648789035870792, −9.341353913055162965908657706357, −8.185882915649647404024575183876, −7.63555118206943388984571425302, −6.52242469678607630504753102602, −5.53972015301862097837353160932, −4.40229502558465218870390475151, −3.26349678920619026469348160229, −1.45281066946860217829463744080, 0.60812656207555306067085475376, 3.15786608490863518431817566907, 3.85543886400834405175648274367, 5.07243960438305366804522719080, 6.31679007014803612186606192828, 7.00344904746531792118800649766, 8.299897228001785872597828110557, 8.923964694202513748391421009786, 10.18334706887135345321171844250, 10.99837834358854903281929938990

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