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Properties

Label 2-480-8.5-c1-0-3
Degree $2$
Conductor $480$
Sign $0.994 - 0.102i$
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 + 3.62·7-s − 9-s + 6.20i·11-s + 0.578i·13-s + 15-s + 1.42·17-s − 5.62i·19-s − 3.62i·21-s + 5.62·23-s − 25-s + i·27-s − 2i·29-s + 2.57·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.447i·5-s + 1.37·7-s − 0.333·9-s + 1.87i·11-s + 0.160i·13-s + 0.258·15-s + 0.344·17-s − 1.29i·19-s − 0.791i·21-s + 1.17·23-s − 0.200·25-s + 0.192i·27-s − 0.371i·29-s + 0.463·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.994 - 0.102i$
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: \((2,\ 480,\ (\ :1/2),\ 0.994 - 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61606 + 0.0828273i\)
\(L(\frac12)\) \(\approx\) \(1.61606 + 0.0828273i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.62T + 7T^{2} \)
11 \( 1 - 6.20iT - 11T^{2} \)
13 \( 1 - 0.578iT - 13T^{2} \)
17 \( 1 - 1.42T + 17T^{2} \)
19 \( 1 + 5.62iT - 19T^{2} \)
23 \( 1 - 5.62T + 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 2.57T + 31T^{2} \)
37 \( 1 - 7.83iT - 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 + 7.25iT - 43T^{2} \)
47 \( 1 + 6.78T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 2.20iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 8.41T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 5.42T + 79T^{2} \)
83 \( 1 - 3.25iT - 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 - 4.84T + 97T^{2} \)
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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

−11.19049434235222689483665113708, −10.18396941509056867777129302756, −9.190460686524497772955679823165, −8.108725486893580732814809167964, −7.30624867890319918299261471385, −6.68807192568426674832427265422, −5.15028590978391376162066916803, −4.46982273018593718076667173879, −2.66201296628114797526670390443, −1.55953267233477966328469179050, 1.22764078304203002998684901058, 3.08865737929044936023117314407, 4.25306373961451252630044210120, 5.31675028718878036081611243001, 5.96819462987063250671690321987, 7.64759894227847194822753314315, 8.399534966623670191255899996439, 8.982194439341265011666650439780, 10.22425884343517412267259219662, 11.09656201129393641596855351269

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