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Properties

Label 2-320-5.2-c4-0-29
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: $\chi_{320} (257, \cdot )$
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

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

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