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Properties

Label 2-480-12.11-c3-0-27
Degree $2$
Conductor $480$
Sign $0.896 - 0.442i$
Analytic cond. $28.3209$
Root an. cond. $5.32174$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.92 + 1.66i)3-s + 5i·5-s − 15.5i·7-s + (21.4 + 16.4i)9-s − 8.50·11-s + 48.3·13-s + (−8.33 + 24.6i)15-s − 33.7i·17-s + 64.9i·19-s + (25.8 − 76.4i)21-s + 159.·23-s − 25·25-s + (78.1 + 116. i)27-s + 108. i·29-s − 215. i·31-s + ⋯
L(s)  = 1  + (0.947 + 0.320i)3-s + 0.447i·5-s − 0.838i·7-s + (0.794 + 0.607i)9-s − 0.233·11-s + 1.03·13-s + (−0.143 + 0.423i)15-s − 0.480i·17-s + 0.784i·19-s + (0.269 − 0.794i)21-s + 1.44·23-s − 0.200·25-s + (0.557 + 0.830i)27-s + 0.695i·29-s − 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.896 - 0.442i$
Analytic conductor: \(28.3209\)
Root analytic conductor: \(5.32174\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :3/2),\ 0.896 - 0.442i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.989734407\)
\(L(\frac12)\) \(\approx\) \(2.989734407\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.92 - 1.66i)T \)
5 \( 1 - 5iT \)
good7 \( 1 + 15.5iT - 343T^{2} \)
11 \( 1 + 8.50T + 1.33e3T^{2} \)
13 \( 1 - 48.3T + 2.19e3T^{2} \)
17 \( 1 + 33.7iT - 4.91e3T^{2} \)
19 \( 1 - 64.9iT - 6.85e3T^{2} \)
23 \( 1 - 159.T + 1.21e4T^{2} \)
29 \( 1 - 108. iT - 2.43e4T^{2} \)
31 \( 1 + 215. iT - 2.97e4T^{2} \)
37 \( 1 - 285.T + 5.06e4T^{2} \)
41 \( 1 - 182. iT - 6.89e4T^{2} \)
43 \( 1 - 13.7iT - 7.95e4T^{2} \)
47 \( 1 - 258.T + 1.03e5T^{2} \)
53 \( 1 - 532. iT - 1.48e5T^{2} \)
59 \( 1 + 475.T + 2.05e5T^{2} \)
61 \( 1 + 342.T + 2.26e5T^{2} \)
67 \( 1 + 280. iT - 3.00e5T^{2} \)
71 \( 1 - 367.T + 3.57e5T^{2} \)
73 \( 1 + 671.T + 3.89e5T^{2} \)
79 \( 1 + 401. iT - 4.93e5T^{2} \)
83 \( 1 - 1.14e3T + 5.71e5T^{2} \)
89 \( 1 - 1.43e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.46e3T + 9.12e5T^{2} \)
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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

−10.67420415765757279383044085443, −9.709015176741658157390821078246, −8.915145047760931574091419298686, −7.85111810824258326487583645640, −7.24942949879267835902706686385, −6.06518921109326933595980300357, −4.62049470697018492993520101170, −3.68325398948966725079657229436, −2.74606509065913149255321129082, −1.19983568758924163032835109966, 1.06071938840141330654730349301, 2.36534462967156323734193784436, 3.42816586874828094362048173062, 4.68356259292977872568172432588, 5.88500272318886030949534717285, 6.91951873599448104766890956190, 8.007600753302809722203641098656, 8.830914916506038019066186428267, 9.194601854613719804556675566107, 10.45256506435237138940672298941

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