Properties

Label 2-320-8.5-c5-0-38
Degree $2$
Conductor $320$
Sign $-0.258 - 0.965i$
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  − 20.6i·3-s + 25i·5-s − 93.5·7-s − 182.·9-s − 659. i·11-s − 183. i·13-s + 515.·15-s − 467.·17-s − 225. i·19-s + 1.92e3i·21-s + 1.10e3·23-s − 625·25-s − 1.24e3i·27-s + 1.73e3i·29-s − 7.27e3·31-s + ⋯
L(s)  = 1  − 1.32i·3-s + 0.447i·5-s − 0.721·7-s − 0.751·9-s − 1.64i·11-s − 0.301i·13-s + 0.591·15-s − 0.391·17-s − 0.143i·19-s + 0.954i·21-s + 0.437·23-s − 0.200·25-s − 0.329i·27-s + 0.383i·29-s − 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.258 - 0.965i$
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: \((2,\ 320,\ (\ :5/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1665308873\)
\(L(\frac12)\) \(\approx\) \(0.1665308873\)
\(L(\frac{7}{2})\) 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 \( 1 + 20.6iT - 243T^{2} \)
7 \( 1 + 93.5T + 1.68e4T^{2} \)
11 \( 1 + 659. iT - 1.61e5T^{2} \)
13 \( 1 + 183. iT - 3.71e5T^{2} \)
17 \( 1 + 467.T + 1.41e6T^{2} \)
19 \( 1 + 225. iT - 2.47e6T^{2} \)
23 \( 1 - 1.10e3T + 6.43e6T^{2} \)
29 \( 1 - 1.73e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.27e3T + 2.86e7T^{2} \)
37 \( 1 - 1.42e4iT - 6.93e7T^{2} \)
41 \( 1 + 4.92e3T + 1.15e8T^{2} \)
43 \( 1 + 1.23e4iT - 1.47e8T^{2} \)
47 \( 1 + 619.T + 2.29e8T^{2} \)
53 \( 1 + 1.69e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.07e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.06e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.19e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.46e4T + 1.80e9T^{2} \)
73 \( 1 + 2.52e4T + 2.07e9T^{2} \)
79 \( 1 - 2.90e4T + 3.07e9T^{2} \)
83 \( 1 - 9.26e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.46e4T + 5.58e9T^{2} \)
97 \( 1 - 4.55e4T + 8.58e9T^{2} \)
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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

−10.28233148866741248073494772250, −8.966146979740868559029339218906, −8.137971796452777236281255363696, −7.05562992831971196871028876536, −6.43420337499020936331062890027, −5.49714548895891574668476670372, −3.56162377428977669839879273814, −2.62254517794785191316548404548, −1.16848387987432076999869294451, −0.04567840241889311487727436727, 1.96594322975377479839435599270, 3.53608099077541312947514437212, 4.43762503147368013867668950277, 5.21365416376646130525510778822, 6.58510079906017574217679376108, 7.67837491259437438679065729685, 9.246852725026220862157252830109, 9.411996337370035299446300228295, 10.35621867690437501557433905887, 11.23395574413496534574535425222

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