Properties

Label 2-320-8.5-c5-0-8
Degree $2$
Conductor $320$
Sign $-0.707 + 0.707i$
Analytic cond. $51.3228$
Root an. cond. $7.16399$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.6i·3-s + 25i·5-s − 164.·7-s − 2.80·9-s + 744. i·11-s + 873. i·13-s − 391.·15-s + 984.·17-s − 1.14e3i·19-s − 2.58e3i·21-s + 578.·23-s − 625·25-s + 3.76e3i·27-s + 5.86e3i·29-s − 9.82e3·31-s + ⋯
L(s)  = 1  + 1.00i·3-s + 0.447i·5-s − 1.27·7-s − 0.0115·9-s + 1.85i·11-s + 1.43i·13-s − 0.449·15-s + 0.826·17-s − 0.730i·19-s − 1.27i·21-s + 0.227·23-s − 0.200·25-s + 0.994i·27-s + 1.29i·29-s − 1.83·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.062844095\)
\(L(\frac12)\) \(\approx\) \(1.062844095\)
\(L(\frac{7}{2})\) 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 \( 1 - 25iT \)
good3 \( 1 - 15.6iT - 243T^{2} \)
7 \( 1 + 164.T + 1.68e4T^{2} \)
11 \( 1 - 744. iT - 1.61e5T^{2} \)
13 \( 1 - 873. iT - 3.71e5T^{2} \)
17 \( 1 - 984.T + 1.41e6T^{2} \)
19 \( 1 + 1.14e3iT - 2.47e6T^{2} \)
23 \( 1 - 578.T + 6.43e6T^{2} \)
29 \( 1 - 5.86e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.82e3T + 2.86e7T^{2} \)
37 \( 1 - 7.88e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.04e4T + 1.15e8T^{2} \)
43 \( 1 + 1.90e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.97e3T + 2.29e8T^{2} \)
53 \( 1 + 3.59e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.93e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.78e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.13e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.39e4T + 1.80e9T^{2} \)
73 \( 1 - 2.27e4T + 2.07e9T^{2} \)
79 \( 1 + 9.47e4T + 3.07e9T^{2} \)
83 \( 1 - 3.47e4iT - 3.93e9T^{2} \)
89 \( 1 + 4.20e4T + 5.58e9T^{2} \)
97 \( 1 + 1.14e4T + 8.58e9T^{2} \)
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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

−11.18498880440374731728349926755, −10.18775227861805161716273723864, −9.619597008922389816433042468458, −9.067650887449205662775179706300, −7.11459520615290772186164991516, −6.87877579931404035735005486786, −5.25825018972571898166424632688, −4.23230973143594276699849673685, −3.35053245874803016550331255476, −1.90376796228761126259659413166, 0.31597524006430416689807754983, 1.06991871032352085552239382970, 2.81432699574371159402605802485, 3.73420903188091062458525203115, 5.79789171653833869056145292308, 5.99883135721734896522147581091, 7.46962487695598356390122019248, 8.103752304825221699579289534890, 9.220651593050585140203681825741, 10.19858134571825913401805602600

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