↑

Properties

Label 2-320-80.29-c1-0-0
Degree $2$
Conductor $320$
Sign $-0.921 - 0.387i$
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  + (−0.734 + 0.734i)3-s + (−1.17 − 1.90i)5-s − 1.71·7-s + 1.92i·9-s + (−2.82 + 2.82i)11-s + (−2.59 + 2.59i)13-s + (2.25 + 0.537i)15-s + 1.89i·17-s + (−2.89 − 2.89i)19-s + (1.25 − 1.25i)21-s − 2.00·23-s + (−2.25 + 4.46i)25-s + (−3.61 − 3.61i)27-s + (−6.72 − 6.72i)29-s + 7.11·31-s + ⋯
L(s)  = 1  + (−0.423 + 0.423i)3-s + (−0.524 − 0.851i)5-s − 0.648·7-s + 0.640i·9-s + (−0.852 + 0.852i)11-s + (−0.719 + 0.719i)13-s + (0.583 + 0.138i)15-s + 0.460i·17-s + (−0.664 − 0.664i)19-s + (0.274 − 0.274i)21-s − 0.418·23-s + (−0.450 + 0.892i)25-s + (−0.695 − 0.695i)27-s + (−1.24 − 1.24i)29-s + 1.27·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.921 - 0.387i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0570875 + 0.283051i\)
\(L(\frac12)\) \(\approx\) \(0.0570875 + 0.283051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 + 1.90i)T \)
good3 \( 1 + (0.734 - 0.734i)T - 3iT^{2} \)
7 \( 1 + 1.71T + 7T^{2} \)
11 \( 1 + (2.82 - 2.82i)T - 11iT^{2} \)
13 \( 1 + (2.59 - 2.59i)T - 13iT^{2} \)
17 \( 1 - 1.89iT - 17T^{2} \)
19 \( 1 + (2.89 + 2.89i)T + 19iT^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 + (6.72 + 6.72i)T + 29iT^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 + (-2.25 - 2.25i)T + 37iT^{2} \)
41 \( 1 - 1.59iT - 41T^{2} \)
43 \( 1 + (-8.06 - 8.06i)T + 43iT^{2} \)
47 \( 1 + 4.43iT - 47T^{2} \)
53 \( 1 + (-0.481 - 0.481i)T + 53iT^{2} \)
59 \( 1 + (3.08 - 3.08i)T - 59iT^{2} \)
61 \( 1 + (-3.46 - 3.46i)T + 61iT^{2} \)
67 \( 1 + (1.80 - 1.80i)T - 67iT^{2} \)
71 \( 1 + 0.379iT - 71T^{2} \)
73 \( 1 - 8.37T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + (-8.24 + 8.24i)T - 83iT^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
97 \( 1 - 6.50iT - 97T^{2} \)
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

−12.00697367385602910890641148544, −11.16798891070024673311291319550, −10.07412953818443109328171134122, −9.456959018872796010772599183630, −8.192030303368132344774634715988, −7.38983945488264303788721635524, −6.03268935831195966919860786345, −4.81683526657843296019461135696, −4.24730731480869555555665123656, −2.32319526533932461385536910979, 0.20308828246496007052498755901, 2.74264371304597787387455333271, 3.73959879610082260847040652376, 5.49518139466418070481460133822, 6.37525346869464467055887917563, 7.29436619328304875702308284783, 8.158240309010356170888724111408, 9.497495985913684696975173738309, 10.48796547938295616710210211918, 11.17647390987075107900921031939

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