Properties

Label 2-480-40.29-c1-0-1
Degree $2$
Conductor $480$
Sign $0.101 - 0.994i$
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  − 3-s + (2.11 + 0.726i)5-s + 4.05i·7-s + 9-s − 0.985i·11-s − 4.94·13-s + (−2.11 − 0.726i)15-s + 4.52i·17-s − 2.60i·19-s − 4.05i·21-s + 3.53i·23-s + (3.94 + 3.07i)25-s − 27-s + 7.59i·29-s + 3.28·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.945 + 0.324i)5-s + 1.53i·7-s + 0.333·9-s − 0.297i·11-s − 1.37·13-s + (−0.546 − 0.187i)15-s + 1.09i·17-s − 0.597i·19-s − 0.885i·21-s + 0.737i·23-s + (0.789 + 0.614i)25-s − 0.192·27-s + 1.41i·29-s + 0.589·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.101 - 0.994i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888620 + 0.802553i\)
\(L(\frac12)\) \(\approx\) \(0.888620 + 0.802553i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + (-2.11 - 0.726i)T \)
good7 \( 1 - 4.05iT - 7T^{2} \)
11 \( 1 + 0.985iT - 11T^{2} \)
13 \( 1 + 4.94T + 13T^{2} \)
17 \( 1 - 4.52iT - 17T^{2} \)
19 \( 1 + 2.60iT - 19T^{2} \)
23 \( 1 - 3.53iT - 23T^{2} \)
29 \( 1 - 7.59iT - 29T^{2} \)
31 \( 1 - 3.28T + 31T^{2} \)
37 \( 1 - 0.945T + 37T^{2} \)
41 \( 1 - 0.568T + 41T^{2} \)
43 \( 1 - 8.45T + 43T^{2} \)
47 \( 1 + 2.60iT - 47T^{2} \)
53 \( 1 + 0.229T + 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 + 8.45T + 67T^{2} \)
71 \( 1 + 1.43T + 71T^{2} \)
73 \( 1 + 11.9iT - 73T^{2} \)
79 \( 1 + 3.28T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−11.17015342001608477511037963396, −10.32255125556292333830278620039, −9.420224361328859620123010372474, −8.765344886641398050947530549274, −7.41841407500620148898153353573, −6.32242678107262605159135288125, −5.63670131398747127111629501554, −4.86058858165686218688797270128, −2.97841773126612876502055257952, −1.91634765867288201033262170352, 0.78857349215925599222326520489, 2.44669549968000715208754961176, 4.27143611009554372742365879392, 4.95931229340575094338685669197, 6.14503190504333502210334240581, 7.08900096895442126730275693544, 7.81067370261997413161506546218, 9.349628622178159898479507360809, 10.02190843560214306704268834139, 10.53042622649115316451913972648

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