↑

Properties

Label 2-360-15.2-c3-0-15
Degree $2$
Conductor $360$
Sign $-0.920 + 0.391i$
Analytic cond. $21.2406$
Root an. cond. $4.60876$
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  + (−10.6 − 3.53i)5-s + (18 + 18i)7-s − 36.7i·11-s + (−51 + 51i)13-s + (25.4 − 25.4i)17-s − 112i·19-s + (36.7 + 36.7i)23-s + (100. + 75i)25-s − 193.·29-s − 284·31-s + (−127. − 254. i)35-s + (−155 − 155i)37-s − 374. i·41-s + (−296 + 296i)43-s + (−209. + 209. i)47-s + ⋯
L(s)  = 1  + (−0.948 − 0.316i)5-s + (0.971 + 0.971i)7-s − 1.00i·11-s + (−1.08 + 1.08i)13-s + (0.363 − 0.363i)17-s − 1.35i·19-s + (0.333 + 0.333i)23-s + (0.800 + 0.599i)25-s − 1.24·29-s − 1.64·31-s + (−0.614 − 1.22i)35-s + (−0.688 − 0.688i)37-s − 1.42i·41-s + (−1.04 + 1.04i)43-s + (−0.649 + 0.649i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.920 + 0.391i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ -0.920 + 0.391i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2824684074\)
\(L(\frac12)\) \(\approx\) \(0.2824684074\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (10.6 + 3.53i)T \)
good7 \( 1 + (-18 - 18i)T + 343iT^{2} \)
11 \( 1 + 36.7iT - 1.33e3T^{2} \)
13 \( 1 + (51 - 51i)T - 2.19e3iT^{2} \)
17 \( 1 + (-25.4 + 25.4i)T - 4.91e3iT^{2} \)
19 \( 1 + 112iT - 6.85e3T^{2} \)
23 \( 1 + (-36.7 - 36.7i)T + 1.21e4iT^{2} \)
29 \( 1 + 193.T + 2.43e4T^{2} \)
31 \( 1 + 284T + 2.97e4T^{2} \)
37 \( 1 + (155 + 155i)T + 5.06e4iT^{2} \)
41 \( 1 + 374. iT - 6.89e4T^{2} \)
43 \( 1 + (296 - 296i)T - 7.95e4iT^{2} \)
47 \( 1 + (209. - 209. i)T - 1.03e5iT^{2} \)
53 \( 1 + (466. + 466. i)T + 1.48e5iT^{2} \)
59 \( 1 + 223.T + 2.05e5T^{2} \)
61 \( 1 - 456T + 2.26e5T^{2} \)
67 \( 1 + (212 + 212i)T + 3.00e5iT^{2} \)
71 \( 1 + 197. iT - 3.57e5T^{2} \)
73 \( 1 + (-105 + 105i)T - 3.89e5iT^{2} \)
79 \( 1 + 340iT - 4.93e5T^{2} \)
83 \( 1 + (461. + 461. i)T + 5.71e5iT^{2} \)
89 \( 1 + 770.T + 7.04e5T^{2} \)
97 \( 1 + (-917 - 917i)T + 9.12e5iT^{2} \)
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

−11.14089395438206264999588594871, −9.309674853274248788769636893673, −8.860459958841605764288991559316, −7.80877754204286259461792350147, −6.99259617145047386070122720029, −5.41753972721266283099486245201, −4.76416314248374887287876123122, −3.38511270905239106685332723626, −1.92818836781794679533588486040, −0.096085046473508951308713917175, 1.63724928184869295439875987200, 3.37997598132996123779009520890, 4.39524229395719293103539845813, 5.35334581619811234450242116350, 7.05573367201857108892181198839, 7.62979823890452787967237593915, 8.265880597260984153461466425600, 9.865803066696260436472335224017, 10.50402416706117857102586471602, 11.35036241253670779956902429288

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