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Properties

Label 2-320-8.5-c5-0-24
Degree $2$
Conductor $320$
Sign $0.965 + 0.258i$
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  + 30.0i·3-s + 25i·5-s − 45.4·7-s − 658.·9-s + 142. i·11-s − 897. i·13-s − 750.·15-s + 11.4·17-s − 1.81e3i·19-s − 1.36e3i·21-s + 3.87e3·23-s − 625·25-s − 1.24e4i·27-s + 3.46e3i·29-s + 1.55e3·31-s + ⋯
L(s)  = 1  + 1.92i·3-s + 0.447i·5-s − 0.350·7-s − 2.70·9-s + 0.355i·11-s − 1.47i·13-s − 0.861·15-s + 0.00964·17-s − 1.15i·19-s − 0.675i·21-s + 1.52·23-s − 0.200·25-s − 3.29i·27-s + 0.764i·29-s + 0.289·31-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(3)\) \(\approx\) \(0.8053894984\)
\(L(\frac12)\) \(\approx\) \(0.8053894984\)
\(L(\frac{7}{2})\) 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 \( 1 - 30.0iT - 243T^{2} \)
7 \( 1 + 45.4T + 1.68e4T^{2} \)
11 \( 1 - 142. iT - 1.61e5T^{2} \)
13 \( 1 + 897. iT - 3.71e5T^{2} \)
17 \( 1 - 11.4T + 1.41e6T^{2} \)
19 \( 1 + 1.81e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.87e3T + 6.43e6T^{2} \)
29 \( 1 - 3.46e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.55e3T + 2.86e7T^{2} \)
37 \( 1 + 9.61e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.48e4T + 1.15e8T^{2} \)
43 \( 1 + 6.32e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.90e4T + 2.29e8T^{2} \)
53 \( 1 - 3.25e4iT - 4.18e8T^{2} \)
59 \( 1 + 652. iT - 7.14e8T^{2} \)
61 \( 1 + 4.71e3iT - 8.44e8T^{2} \)
67 \( 1 + 4.63e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.26e4T + 1.80e9T^{2} \)
73 \( 1 - 6.93e4T + 2.07e9T^{2} \)
79 \( 1 + 3.32e4T + 3.07e9T^{2} \)
83 \( 1 - 5.53e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.32e4T + 5.58e9T^{2} \)
97 \( 1 + 8.24e4T + 8.58e9T^{2} \)
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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

−10.78459788516088302102460218644, −9.855716759444342457562894965403, −9.196235864048216673752106511959, −8.199753088609583648875760955249, −6.76274081925020163829234050800, −5.42320508722180309578892333334, −4.78964704634846752073375857993, −3.44210176512020067250681333969, −2.85497455087869995988478517833, −0.23147502534738586381053198556, 1.09544435157951971165694200081, 1.95643298888955788323731527200, 3.30611888647509315617046364118, 5.09231404292119390841156581880, 6.38051165862508101558152118271, 6.78298866115021272155255383283, 7.997983653720348226107087846980, 8.610704112819811956343702863466, 9.716677032777670624889652175948, 11.40094848918861779479341097528

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