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Properties

Label 2-480-5.4-c1-0-4
Degree $2$
Conductor $480$
Sign $0.447 - 0.894i$
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 + (1 − 2i)5-s + 4i·7-s − 9-s + 4i·13-s + (2 + i)15-s + 8·19-s − 4·21-s + 4i·23-s + (−3 − 4i)25-s − i·27-s + 6·29-s − 8·31-s + (8 + 4i)35-s + 4i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.447 − 0.894i)5-s + 1.51i·7-s − 0.333·9-s + 1.10i·13-s + (0.516 + 0.258i)15-s + 1.83·19-s − 0.872·21-s + 0.834i·23-s + (−0.600 − 0.800i)25-s − 0.192i·27-s + 1.11·29-s − 1.43·31-s + (1.35 + 0.676i)35-s + 0.657i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26467 + 0.781614i\)
\(L(\frac12)\) \(\approx\) \(1.26467 + 0.781614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−11.41743970778774975400303966308, −9.929820598187428319125980332286, −9.251093571579624393858827862030, −8.861824877114813117932923518761, −7.69100625337343659531140064385, −6.19522811229739267154054994597, −5.40168999533577827146192711099, −4.66032129314467812190984027118, −3.15891568248029124408836802439, −1.75732140595578105415486323736, 0.999355934092705003166772031466, 2.74322786618082339973145301394, 3.76388813856350738805375389722, 5.29939454441125981195058243999, 6.33194324103217618765407995558, 7.42734279973460185432009308593, 7.58046593760750272291590356017, 9.121682277424501562838310149212, 10.31691619638260937462241075078, 10.57755576727606547849543533034

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