↑

Properties

Label 2-320-40.29-c5-0-29
Degree $2$
Conductor $320$
Sign $0.357 - 0.933i$
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  − 28.5·3-s + (−5.81 + 55.5i)5-s + 92.3i·7-s + 571.·9-s + 541. i·11-s + 782.·13-s + (165. − 1.58e3i)15-s − 1.55e3i·17-s + 484. i·19-s − 2.63e3i·21-s − 1.09e3i·23-s + (−3.05e3 − 646. i)25-s − 9.37e3·27-s − 597. i·29-s + 8.86e3·31-s + ⋯
L(s)  = 1  − 1.83·3-s + (−0.104 + 0.994i)5-s + 0.712i·7-s + 2.35·9-s + 1.34i·11-s + 1.28·13-s + (0.190 − 1.82i)15-s − 1.30i·17-s + 0.307i·19-s − 1.30i·21-s − 0.432i·23-s + (−0.978 − 0.206i)25-s − 2.47·27-s − 0.131i·29-s + 1.65·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.357 - 0.933i$
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: \((2,\ 320,\ (\ :5/2),\ 0.357 - 0.933i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.178255591\)
\(L(\frac12)\) \(\approx\) \(1.178255591\)
\(L(\frac{7}{2})\) 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 \( 1 + (5.81 - 55.5i)T \)
good3 \( 1 + 28.5T + 243T^{2} \)
7 \( 1 - 92.3iT - 1.68e4T^{2} \)
11 \( 1 - 541. iT - 1.61e5T^{2} \)
13 \( 1 - 782.T + 3.71e5T^{2} \)
17 \( 1 + 1.55e3iT - 1.41e6T^{2} \)
19 \( 1 - 484. iT - 2.47e6T^{2} \)
23 \( 1 + 1.09e3iT - 6.43e6T^{2} \)
29 \( 1 + 597. iT - 2.05e7T^{2} \)
31 \( 1 - 8.86e3T + 2.86e7T^{2} \)
37 \( 1 - 1.36e4T + 6.93e7T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 + 1.82e3T + 1.47e8T^{2} \)
47 \( 1 + 1.79e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.65e3T + 4.18e8T^{2} \)
59 \( 1 + 1.44e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.45e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.68e4T + 1.35e9T^{2} \)
71 \( 1 - 4.80e4T + 1.80e9T^{2} \)
73 \( 1 - 6.06e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.81e4T + 3.07e9T^{2} \)
83 \( 1 - 2.45e4T + 3.93e9T^{2} \)
89 \( 1 - 1.17e5T + 5.58e9T^{2} \)
97 \( 1 - 3.77e4iT - 8.58e9T^{2} \)
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

−11.15847066230491538442325895299, −10.25538954063385179380770466852, −9.518139471736587956965369136771, −7.80208869517816908256767654650, −6.73726598383257128227104366903, −6.21009862337512040257131561419, −5.16439649180927758348644983750, −4.13045530191499530450879607279, −2.37019538467195793629830930018, −0.789386651532782260796325753724, 0.74190205412005657334105676040, 1.12960743037922320744119293106, 3.84833922434319025411829689884, 4.64923879097424342405154247116, 6.02536767910065945974712826763, 6.09354581733175853529093065994, 7.70375600790689082836613161595, 8.695221882264239209399399769324, 9.984611988610378845401937282661, 11.00192003393488386309829141220

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