↑

Properties

Label 2-320-40.29-c5-0-10
Degree $2$
Conductor $320$
Sign $-0.883 - 0.468i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 24.8·3-s + (−53.4 + 16.3i)5-s + 100. i·7-s + 372.·9-s + 5.36i·11-s − 506.·13-s + (−1.32e3 + 406. i)15-s + 1.45e3i·17-s − 722. i·19-s + 2.48e3i·21-s − 3.08e3i·23-s + (2.58e3 − 1.75e3i)25-s + 3.20e3·27-s + 7.29e3i·29-s − 8.60e3·31-s + ⋯
L(s)  = 1  + 1.59·3-s + (−0.956 + 0.293i)5-s + 0.773i·7-s + 1.53·9-s + 0.0133i·11-s − 0.831·13-s + (−1.52 + 0.466i)15-s + 1.22i·17-s − 0.459i·19-s + 1.23i·21-s − 1.21i·23-s + (0.828 − 0.560i)25-s + 0.846·27-s + 1.61i·29-s − 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.883 - 0.468i$
Analytic conductor: \(51.3228\)
Root analytic conductor: \(7.16399\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :5/2),\ -0.883 - 0.468i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.352144581\)
\(L(\frac12)\) \(\approx\) \(1.352144581\)
\(L(\frac{7}{2})\) 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 \( 1 + (53.4 - 16.3i)T \)
good3 \( 1 - 24.8T + 243T^{2} \)
7 \( 1 - 100. iT - 1.68e4T^{2} \)
11 \( 1 - 5.36iT - 1.61e5T^{2} \)
13 \( 1 + 506.T + 3.71e5T^{2} \)
17 \( 1 - 1.45e3iT - 1.41e6T^{2} \)
19 \( 1 + 722. iT - 2.47e6T^{2} \)
23 \( 1 + 3.08e3iT - 6.43e6T^{2} \)
29 \( 1 - 7.29e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.60e3T + 2.86e7T^{2} \)
37 \( 1 + 5.27e3T + 6.93e7T^{2} \)
41 \( 1 + 1.81e4T + 1.15e8T^{2} \)
43 \( 1 + 8.34e3T + 1.47e8T^{2} \)
47 \( 1 - 2.21e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.33e4T + 4.18e8T^{2} \)
59 \( 1 + 1.09e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.09e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.30e4T + 1.35e9T^{2} \)
71 \( 1 - 3.60e4T + 1.80e9T^{2} \)
73 \( 1 - 6.42e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.86e4T + 3.07e9T^{2} \)
83 \( 1 + 4.97e4T + 3.93e9T^{2} \)
89 \( 1 - 7.76e3T + 5.58e9T^{2} \)
97 \( 1 + 3.68e4iT - 8.58e9T^{2} \)
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

−11.11653928455750449962273028069, −10.13602329441708379858529932508, −8.931892098140783734543736944562, −8.517914899669658559262882371991, −7.59411710147936331520477105836, −6.70068108773810657900065193894, −4.97011244680936673503771751045, −3.74204636379407217170905185573, −2.92320367427630662931586514785, −1.86325758900006500940121914949, 0.26080891014079971409091142885, 1.86374653166404933977668404552, 3.23855615431135178203301253527, 3.91871329216541684689077206338, 5.07412183372687089276033574018, 7.16906083555839657214078596318, 7.53875010547911849522648949770, 8.432452017541406878608645521286, 9.368610933860315211338448681592, 10.10170749390243935522057374490

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