↑

Properties

Label 2-336-21.5-c3-0-36
Degree $2$
Conductor $336$
Sign $-0.998 + 0.0592i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
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  + (−4.88 − 1.76i)3-s + (6.04 − 10.4i)5-s + (−15.9 + 9.39i)7-s + (20.7 + 17.2i)9-s + (30.2 − 17.4i)11-s − 81.1i·13-s + (−48.0 + 40.5i)15-s + (4.14 + 7.17i)17-s + (52.7 + 30.4i)19-s + (94.5 − 17.7i)21-s + (−157. − 90.7i)23-s + (−10.6 − 18.4i)25-s + (−71.3 − 120. i)27-s + 75.6i·29-s + (−77.3 + 44.6i)31-s + ⋯
L(s)  = 1  + (−0.940 − 0.338i)3-s + (0.540 − 0.936i)5-s + (−0.861 + 0.507i)7-s + (0.770 + 0.637i)9-s + (0.829 − 0.479i)11-s − 1.73i·13-s + (−0.826 + 0.698i)15-s + (0.0591 + 0.102i)17-s + (0.636 + 0.367i)19-s + (0.982 − 0.184i)21-s + (−1.42 − 0.822i)23-s + (−0.0850 − 0.147i)25-s + (−0.508 − 0.860i)27-s + 0.484i·29-s + (−0.448 + 0.258i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0592i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.998 + 0.0592i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.998 + 0.0592i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6125065659\)
\(L(\frac12)\) \(\approx\) \(0.6125065659\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.88 + 1.76i)T \)
7 \( 1 + (15.9 - 9.39i)T \)
good5 \( 1 + (-6.04 + 10.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.2 + 17.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 81.1iT - 2.19e3T^{2} \)
17 \( 1 + (-4.14 - 7.17i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-52.7 - 30.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (157. + 90.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 75.6iT - 2.43e4T^{2} \)
31 \( 1 + (77.3 - 44.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (58.4 - 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 147.T + 6.89e4T^{2} \)
43 \( 1 + 389.T + 7.95e4T^{2} \)
47 \( 1 + (-99.8 + 172. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (578. - 333. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-8.54 - 14.8i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (661. + 381. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (428. + 741. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 700. iT - 3.57e5T^{2} \)
73 \( 1 + (748. - 432. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-463. + 802. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.05e3T + 5.71e5T^{2} \)
89 \( 1 + (-224. + 389. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 213. iT - 9.12e5T^{2} \)
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

−10.56363981218700979539666107826, −9.874024757719977749037662360337, −8.843583083280888449963567417814, −7.83467222390311013519719883415, −6.42022676762978334763818680403, −5.79257923383735820519300560043, −4.97377623049680358315006834706, −3.34268230657463353500764439264, −1.51956579570652740850530498158, −0.25032805820467475939890398100, 1.73384890161862811878831046036, 3.53306013630386502717743338943, 4.48675221873336271271449991730, 6.01815975768998395300751029801, 6.62869659918719790598661199661, 7.29844572566571656118728871187, 9.366619267172233323348905450433, 9.709020974181152670150027732900, 10.60943406273416972814718243769, 11.60594613813089630956964759993

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