Properties

Label 2-320-8.5-c1-0-2
Degree $2$
Conductor $320$
Sign $-0.258 - 0.965i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.73i·3-s + i·5-s + 4.73·7-s − 4.46·9-s + 3.46i·11-s − 3.46i·13-s − 2.73·15-s − 3.46·17-s − 2i·19-s + 12.9i·21-s − 2.19·23-s − 25-s − 3.99i·27-s − 2.53·31-s − 9.46·33-s + ⋯
L(s)  = 1  + 1.57i·3-s + 0.447i·5-s + 1.78·7-s − 1.48·9-s + 1.04i·11-s − 0.960i·13-s − 0.705·15-s − 0.840·17-s − 0.458i·19-s + 2.82i·21-s − 0.457·23-s − 0.200·25-s − 0.769i·27-s − 0.455·31-s − 1.64·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891812 + 1.16223i\)
\(L(\frac12)\) \(\approx\) \(0.891812 + 1.16223i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 - 2.73iT - 3T^{2} \)
7 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 2.19T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 9.46T + 41T^{2} \)
43 \( 1 + 0.196iT - 43T^{2} \)
47 \( 1 - 2.19T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 0.928iT - 61T^{2} \)
67 \( 1 + 0.196iT - 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 - 6.39T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 1.26iT - 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−11.41252855123899983986942115017, −10.93074474883918805510494030016, −10.19976779368031626323688120447, −9.238621253436707852103916216737, −8.260522407781695279026341291730, −7.28482561540555240027552186332, −5.56678722007914872470340498459, −4.74923742536354138952265870981, −3.97851902355099256593992072877, −2.31690934466756140279194415739, 1.22447303076733250374926377475, 2.20992760395063336859024110249, 4.30954755324240951155302494286, 5.55839792909619361598525833106, 6.57404807877663760465410882540, 7.71261604603941745553429114516, 8.262800374752680871730734032005, 9.070321189516178084666430431450, 10.91456290682788660469075623065, 11.51619100062475453597768216747

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