↑

Properties

Label 2-320-80.43-c1-0-8
Degree $2$
Conductor $320$
Sign $-0.489 + 0.872i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.85i·3-s + (1.43 − 1.71i)5-s + (0.458 − 0.458i)7-s − 5.15·9-s + (0.492 − 0.492i)11-s + 4.52·13-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 + 6.15i·27-s + (−3.83 − 3.83i)29-s + 0.139i·31-s + ⋯
L(s)  = 1  − 1.64i·3-s + (0.641 − 0.766i)5-s + (0.173 − 0.173i)7-s − 1.71·9-s + (0.148 − 0.148i)11-s + 1.25·13-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 + 1.18i·27-s + (−0.712 − 0.712i)29-s + 0.0251i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.489 + 0.872i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.489 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723909 - 1.23590i\)
\(L(\frac12)\) \(\approx\) \(0.723909 - 1.23590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 + 1.71i)T \)
good3 \( 1 + 2.85iT - 3T^{2} \)
7 \( 1 + (-0.458 + 0.458i)T - 7iT^{2} \)
11 \( 1 + (-0.492 + 0.492i)T - 11iT^{2} \)
13 \( 1 - 4.52T + 13T^{2} \)
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 \( 1 - 5.84T + 37T^{2} \)
41 \( 1 + 4.55iT - 41T^{2} \)
43 \( 1 - 7.49T + 43T^{2} \)
47 \( 1 + (-4.14 - 4.14i)T + 47iT^{2} \)
53 \( 1 - 2.75iT - 53T^{2} \)
59 \( 1 + (-3.62 - 3.62i)T + 59iT^{2} \)
61 \( 1 + (-3.72 + 3.72i)T - 61iT^{2} \)
67 \( 1 + 3.32T + 67T^{2} \)
71 \( 1 + 1.37T + 71T^{2} \)
73 \( 1 + (2.55 - 2.55i)T - 73iT^{2} \)
79 \( 1 - 3.86T + 79T^{2} \)
83 \( 1 + 14.4iT - 83T^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 + (4.95 - 4.95i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.49962619813389835042902918747, −10.59896420311378552202037207197, −9.096340465756719301539958190376, −8.403633898522118068714768499615, −7.55571575617892834233668696559, −6.21712436762712080805528214656, −5.90743465097336226834598707979, −4.11717367505288773775145466816, −2.16364433750475245635072702046, −1.15691674805094895955041589294, 2.55607753578272133889128067769, 3.77694127437807683075671061909, 4.82312335514437094378315460686, 5.89769567397180310714928320077, 6.93063844041708132759195710780, 8.650543348379038583896513009085, 9.226276580480759236926900815370, 10.12865493860614398879064003505, 11.11622626789582310487609703943, 11.19422383153184914658881380708

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