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Properties

Label 2-320-80.3-c1-0-1
Degree $2$
Conductor $320$
Sign $0.406 - 0.913i$
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  − 2.85·3-s + (−1.71 − 1.43i)5-s + (0.458 − 0.458i)7-s + 5.15·9-s + (0.492 + 0.492i)11-s + 4.52i·13-s + (4.89 + 4.09i)15-s + (−3.12 + 3.12i)17-s + (4.04 + 4.04i)19-s + (−1.31 + 1.31i)21-s + (1.80 + 1.80i)23-s + (0.881 + 4.92i)25-s − 6.15·27-s + (3.83 − 3.83i)29-s − 0.139i·31-s + ⋯
L(s)  = 1  − 1.64·3-s + (−0.766 − 0.641i)5-s + (0.173 − 0.173i)7-s + 1.71·9-s + (0.148 + 0.148i)11-s + 1.25i·13-s + (1.26 + 1.05i)15-s + (−0.758 + 0.758i)17-s + (0.928 + 0.928i)19-s + (−0.285 + 0.285i)21-s + (0.376 + 0.376i)23-s + (0.176 + 0.984i)25-s − 1.18·27-s + (0.712 − 0.712i)29-s − 0.0251i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.406 - 0.913i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.406 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.441055 + 0.286445i\)
\(L(\frac12)\) \(\approx\) \(0.441055 + 0.286445i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 1.43i)T \)
good3 \( 1 + 2.85T + 3T^{2} \)
7 \( 1 + (-0.458 + 0.458i)T - 7iT^{2} \)
11 \( 1 + (-0.492 - 0.492i)T + 11iT^{2} \)
13 \( 1 - 4.52iT - 13T^{2} \)
17 \( 1 + (3.12 - 3.12i)T - 17iT^{2} \)
19 \( 1 + (-4.04 - 4.04i)T + 19iT^{2} \)
23 \( 1 + (-1.80 - 1.80i)T + 23iT^{2} \)
29 \( 1 + (-3.83 + 3.83i)T - 29iT^{2} \)
31 \( 1 + 0.139iT - 31T^{2} \)
37 \( 1 + 5.84iT - 37T^{2} \)
41 \( 1 - 4.55iT - 41T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 + (4.14 + 4.14i)T + 47iT^{2} \)
53 \( 1 - 2.75T + 53T^{2} \)
59 \( 1 + (3.62 - 3.62i)T - 59iT^{2} \)
61 \( 1 + (-3.72 - 3.72i)T + 61iT^{2} \)
67 \( 1 - 3.32iT - 67T^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 + (-2.55 + 2.55i)T - 73iT^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 - 3.35T + 89T^{2} \)
97 \( 1 + (4.95 - 4.95i)T - 97iT^{2} \)
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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

−11.64612099640019055187535201738, −11.26280945285724195669065826901, −10.15910320955914708777609881374, −9.086683052475929105101355511753, −7.83031848162192838044397874622, −6.82013059976632830368081012645, −5.87106353984743004618608101517, −4.75282345520827768756866469501, −4.04076040260219058660963308265, −1.31442909494607635258817461871, 0.53109396904516752783415325480, 3.06435025768709554487854608556, 4.65837953525682763546512066517, 5.43539546981367348649872829848, 6.64806145970344842548145300682, 7.25907832311513520379697272328, 8.546942614826057481767489031156, 10.00793237529205675998535368965, 10.83966559911533212440950789209, 11.40105762930080063909809228524

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