↑

Properties

Label 2-460-5.4-c3-0-25
Degree $2$
Conductor $460$
Sign $0.170 + 0.985i$
Analytic cond. $27.1408$
Root an. cond. $5.20969$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.97i·3-s + (11.0 − 1.90i)5-s − 13.0i·7-s + 23.1·9-s − 24.3·11-s − 74.8i·13-s + (3.76 + 21.7i)15-s − 65.9i·17-s − 112.·19-s + 25.7·21-s + 23i·23-s + (117. − 42.0i)25-s + 98.9i·27-s − 105.·29-s + 20.5·31-s + ⋯
L(s)  = 1  + 0.379i·3-s + (0.985 − 0.170i)5-s − 0.703i·7-s + 0.855·9-s − 0.666·11-s − 1.59i·13-s + (0.0648 + 0.374i)15-s − 0.940i·17-s − 1.35·19-s + 0.267·21-s + 0.208i·23-s + (0.941 − 0.336i)25-s + 0.705i·27-s − 0.673·29-s + 0.118·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.170 + 0.985i$
Analytic conductor: \(27.1408\)
Root analytic conductor: \(5.20969\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :3/2),\ 0.170 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.948394326\)
\(L(\frac12)\) \(\approx\) \(1.948394326\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.0 + 1.90i)T \)
23 \( 1 - 23iT \)
good3 \( 1 - 1.97iT - 27T^{2} \)
7 \( 1 + 13.0iT - 343T^{2} \)
11 \( 1 + 24.3T + 1.33e3T^{2} \)
13 \( 1 + 74.8iT - 2.19e3T^{2} \)
17 \( 1 + 65.9iT - 4.91e3T^{2} \)
19 \( 1 + 112.T + 6.85e3T^{2} \)
29 \( 1 + 105.T + 2.43e4T^{2} \)
31 \( 1 - 20.5T + 2.97e4T^{2} \)
37 \( 1 + 227. iT - 5.06e4T^{2} \)
41 \( 1 + 510.T + 6.89e4T^{2} \)
43 \( 1 - 339. iT - 7.95e4T^{2} \)
47 \( 1 + 538. iT - 1.03e5T^{2} \)
53 \( 1 - 47.8iT - 1.48e5T^{2} \)
59 \( 1 - 407.T + 2.05e5T^{2} \)
61 \( 1 - 373.T + 2.26e5T^{2} \)
67 \( 1 - 191. iT - 3.00e5T^{2} \)
71 \( 1 - 380.T + 3.57e5T^{2} \)
73 \( 1 + 1.22e3iT - 3.89e5T^{2} \)
79 \( 1 - 572.T + 4.93e5T^{2} \)
83 \( 1 + 394. iT - 5.71e5T^{2} \)
89 \( 1 - 881.T + 7.04e5T^{2} \)
97 \( 1 - 1.17e3iT - 9.12e5T^{2} \)
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

−10.29444999370318303820105939505, −9.859943509567186684981984477043, −8.723866809013058679190833471981, −7.66319733484827246506766268412, −6.74637674459562890302718632017, −5.52344937114434070116285552445, −4.78378610791859167892653313062, −3.48500316473026293361905207107, −2.12089931517584303136026019648, −0.59708683852983738858287363968, 1.67079675910129520835518948032, 2.32509193552714113136217385601, 4.06887026706900924077301573438, 5.22597800809245304143476663857, 6.37470516676243430102987589460, 6.85268960141393010343756837371, 8.241577939991137410102537929802, 9.050620714595344257073042093524, 9.980910900910331753335420521613, 10.68990194621266240934417444978

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