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Properties

Label 2-480-40.29-c1-0-9
Degree $2$
Conductor $480$
Sign $0.530 + 0.847i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.530 + 0.847i$
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: \((2,\ 480,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27706 - 0.706889i\)
\(L(\frac12)\) \(\approx\) \(1.27706 - 0.706889i\)
\(L(\frac{3}{2})\) 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

−10.89872432101918738483299629957, −10.11789302140291410865896650915, −8.861017395279788473925729750022, −8.132474107587737710862591966870, −7.18484783892715893220936665610, −6.61955037002223859956312775688, −4.79601001560153968163757467580, −3.84981345762096773235078596995, −3.06800677854131571684795347737, −0.919617039958997677208070014175, 1.78018408387523595314253450485, 3.25702569882354700098196248490, 4.18008633837503964623073510671, 5.52535646806503124228940188114, 6.44630198777069939182736022673, 7.911490082913086618203997188597, 8.397914573364896415064845485984, 9.079440800244020319266845179690, 10.15469986166632064698444129873, 11.37023398805462400895477677125

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