Properties

Label 2-320-5.4-c5-0-11
Degree $2$
Conductor $320$
Sign $-0.335 - 0.942i$
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  + 21.9i·3-s + (−18.7 − 52.6i)5-s − 29.6i·7-s − 239.·9-s − 227.·11-s − 1.06e3i·13-s + (1.15e3 − 412. i)15-s + 686. i·17-s + 1.30e3·19-s + 652.·21-s + 4.07e3i·23-s + (−2.42e3 + 1.97e3i)25-s + 78.9i·27-s − 4.53e3·29-s + 7.69e3·31-s + ⋯
L(s)  = 1  + 1.40i·3-s + (−0.335 − 0.942i)5-s − 0.229i·7-s − 0.985·9-s − 0.565·11-s − 1.75i·13-s + (1.32 − 0.472i)15-s + 0.575i·17-s + 0.831·19-s + 0.322·21-s + 1.60i·23-s + (−0.774 + 0.632i)25-s + 0.0208i·27-s − 1.00·29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.335 - 0.942i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -0.335 - 0.942i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.359814293\)
\(L(\frac12)\) \(\approx\) \(1.359814293\)
\(L(\frac{7}{2})\) 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 \( 1 + (18.7 + 52.6i)T \)
good3 \( 1 - 21.9iT - 243T^{2} \)
7 \( 1 + 29.6iT - 1.68e4T^{2} \)
11 \( 1 + 227.T + 1.61e5T^{2} \)
13 \( 1 + 1.06e3iT - 3.71e5T^{2} \)
17 \( 1 - 686. iT - 1.41e6T^{2} \)
19 \( 1 - 1.30e3T + 2.47e6T^{2} \)
23 \( 1 - 4.07e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.53e3T + 2.05e7T^{2} \)
31 \( 1 - 7.69e3T + 2.86e7T^{2} \)
37 \( 1 - 9.32e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.75e3T + 1.15e8T^{2} \)
43 \( 1 + 2.93e3iT - 1.47e8T^{2} \)
47 \( 1 - 3.69e3iT - 2.29e8T^{2} \)
53 \( 1 - 4.81e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.11e4T + 7.14e8T^{2} \)
61 \( 1 - 3.38e4T + 8.44e8T^{2} \)
67 \( 1 + 2.48e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.41e4T + 1.80e9T^{2} \)
73 \( 1 - 7.31e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.36e4T + 3.07e9T^{2} \)
83 \( 1 - 6.73e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.02e4T + 5.58e9T^{2} \)
97 \( 1 + 1.54e4iT - 8.58e9T^{2} \)
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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

−10.88108240966610785837591813777, −10.06438387935245141650596974431, −9.436315578366650945270819206516, −8.301773778763766033853285946303, −7.61355287826379897831350023207, −5.62582915816638836306070354192, −5.12672795000023313068747992610, −4.00031484833398646578350782847, −3.11658724230920457438050401461, −1.01372085696832029790028759382, 0.42313082274873531388509556507, 1.95831310449172689693424778210, 2.78569555321373348121955451114, 4.36352416833875110236675274820, 5.96169557545065336554136775350, 6.85810097199802927647328637311, 7.36366622649597570499223713278, 8.365479642574921734625225810991, 9.519523312229056898590742470514, 10.74451935837940428058878165525

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