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Properties

Label 2-320-5.3-c4-0-13
Degree $2$
Conductor $320$
Sign $0.995 - 0.0898i$
Analytic cond. $33.0783$
Root an. cond. $5.75138$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9 − 9i)3-s + (15 + 20i)5-s + (29 − 29i)7-s + 81i·9-s + 118·11-s + (−69 − 69i)13-s + (45 − 315i)15-s + (−271 + 271i)17-s + 280i·19-s − 522·21-s + (269 + 269i)23-s + (−175 + 600i)25-s + 680i·29-s + 202·31-s + (−1.06e3 − 1.06e3i)33-s + ⋯
L(s)  = 1  + (−1 − i)3-s + (0.599 + 0.800i)5-s + (0.591 − 0.591i)7-s + i·9-s + 0.975·11-s + (−0.408 − 0.408i)13-s + (0.200 − 1.39i)15-s + (−0.937 + 0.937i)17-s + 0.775i·19-s − 1.18·21-s + (0.508 + 0.508i)23-s + (−0.280 + 0.960i)25-s + 0.808i·29-s + 0.210·31-s + (−0.975 − 0.975i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 - 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.995 - 0.0898i$
Analytic conductor: \(33.0783\)
Root analytic conductor: \(5.75138\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :2),\ 0.995 - 0.0898i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.562377393\)
\(L(\frac12)\) \(\approx\) \(1.562377393\)
\(L(3)\) 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 \( 1 + (-15 - 20i)T \)
good3 \( 1 + (9 + 9i)T + 81iT^{2} \)
7 \( 1 + (-29 + 29i)T - 2.40e3iT^{2} \)
11 \( 1 - 118T + 1.46e4T^{2} \)
13 \( 1 + (69 + 69i)T + 2.85e4iT^{2} \)
17 \( 1 + (271 - 271i)T - 8.35e4iT^{2} \)
19 \( 1 - 280iT - 1.30e5T^{2} \)
23 \( 1 + (-269 - 269i)T + 2.79e5iT^{2} \)
29 \( 1 - 680iT - 7.07e5T^{2} \)
31 \( 1 - 202T + 9.23e5T^{2} \)
37 \( 1 + (-651 + 651i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.68e3T + 2.82e6T^{2} \)
43 \( 1 + (1.08e3 + 1.08e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.26e3 + 1.26e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-611 - 611i)T + 7.89e6iT^{2} \)
59 \( 1 - 1.16e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.59e3T + 1.38e7T^{2} \)
67 \( 1 + (-751 + 751i)T - 2.01e7iT^{2} \)
71 \( 1 - 6.44e3T + 2.54e7T^{2} \)
73 \( 1 + (2.95e3 + 2.95e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.05e4iT - 3.89e7T^{2} \)
83 \( 1 + (-6.23e3 - 6.23e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.44e4iT - 6.27e7T^{2} \)
97 \( 1 + (7.31e3 - 7.31e3i)T - 8.85e7iT^{2} \)
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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

−11.01762148796394526213899737925, −10.46577864984776010251001490509, −9.194830845253175111750248185477, −7.79759814856610025795185698188, −6.94497326190464113141106216272, −6.29016973021908785390084376222, −5.34050603876604129354937448069, −3.82981083777102127284975139834, −2.03648226993747861063418503083, −1.03600447697540845386744642628, 0.66429408674969750247922005587, 2.29772800655352933009252419353, 4.38902995179078359603608706702, 4.82960546435574340720827501729, 5.81214576845015926630218461563, 6.80808687858650603736965031815, 8.504438475767380224086509619594, 9.318955978709490159515200194133, 9.895740591167785603047595924873, 11.30246925645301388823958042361

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