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Properties

Label 2-320-5.4-c7-0-47
Degree $2$
Conductor $320$
Sign $-0.0534 - 0.998i$
Analytic cond. $99.9632$
Root an. cond. $9.99816$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 52.5i·3-s + (14.9 + 279. i)5-s − 338. i·7-s − 572.·9-s + 3.02e3·11-s − 2.45e3i·13-s + (−1.46e4 + 785. i)15-s + 8.49e3i·17-s + 4.32e4·19-s + 1.78e4·21-s − 4.45e4i·23-s + (−7.76e4 + 8.34e3i)25-s + 8.48e4i·27-s + 1.29e5·29-s + 2.04e5·31-s + ⋯
L(s)  = 1  + 1.12i·3-s + (0.0534 + 0.998i)5-s − 0.373i·7-s − 0.261·9-s + 0.684·11-s − 0.310i·13-s + (−1.12 + 0.0600i)15-s + 0.419i·17-s + 1.44·19-s + 0.419·21-s − 0.763i·23-s + (−0.994 + 0.106i)25-s + 0.829i·27-s + 0.987·29-s + 1.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0534 - 0.998i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0534 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.0534 - 0.998i$
Analytic conductor: \(99.9632\)
Root analytic conductor: \(9.99816\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :7/2),\ -0.0534 - 0.998i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.749417701\)
\(L(\frac12)\) \(\approx\) \(2.749417701\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-14.9 - 279. i)T \)
good3 \( 1 - 52.5iT - 2.18e3T^{2} \)
7 \( 1 + 338. iT - 8.23e5T^{2} \)
11 \( 1 - 3.02e3T + 1.94e7T^{2} \)
13 \( 1 + 2.45e3iT - 6.27e7T^{2} \)
17 \( 1 - 8.49e3iT - 4.10e8T^{2} \)
19 \( 1 - 4.32e4T + 8.93e8T^{2} \)
23 \( 1 + 4.45e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.29e5T + 1.72e10T^{2} \)
31 \( 1 - 2.04e5T + 2.75e10T^{2} \)
37 \( 1 + 4.24e5iT - 9.49e10T^{2} \)
41 \( 1 - 8.33e5T + 1.94e11T^{2} \)
43 \( 1 - 6.21e5iT - 2.71e11T^{2} \)
47 \( 1 + 8.95e5iT - 5.06e11T^{2} \)
53 \( 1 - 5.08e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.29e4T + 2.48e12T^{2} \)
61 \( 1 - 2.51e6T + 3.14e12T^{2} \)
67 \( 1 - 1.41e6iT - 6.06e12T^{2} \)
71 \( 1 + 3.21e6T + 9.09e12T^{2} \)
73 \( 1 + 6.52e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.53e6T + 1.92e13T^{2} \)
83 \( 1 - 3.83e6iT - 2.71e13T^{2} \)
89 \( 1 - 6.52e6T + 4.42e13T^{2} \)
97 \( 1 - 3.16e6iT - 8.07e13T^{2} \)
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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

−10.49610199429303387734026826676, −9.942794267706397089090480717595, −9.068588892505857400654634139573, −7.76239885056150860127529261173, −6.80212481865425158492491362650, −5.75110890477618196264814181648, −4.46832251849066558220983456648, −3.66604852304094252073673278640, −2.65437386188610673240077248254, −0.940075784854250444809683392104, 0.836607769275188071800291468701, 1.36234705649558117268540218896, 2.66275766222845806574147838714, 4.22422879238924896656159553027, 5.35303461202990275915429539663, 6.36980092659997886090210429099, 7.35081488714279710100003588834, 8.205800379295689863062236410685, 9.164021715289086224147252007273, 9.933207094608792602260647580512

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