Properties

Label 2-336-21.5-c3-0-15
Degree $2$
Conductor $336$
Sign $0.633 - 0.773i$
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  + (2.24 − 4.68i)3-s + (−5.80 + 10.0i)5-s + (18.4 + 2.09i)7-s + (−16.9 − 21.0i)9-s + (−15.5 + 8.95i)11-s + 62.4i·13-s + (34.1 + 49.7i)15-s + (10.7 + 18.5i)17-s + (−9.50 − 5.48i)19-s + (51.0 − 81.5i)21-s + (59.8 + 34.5i)23-s + (−4.82 − 8.35i)25-s + (−136. + 32.3i)27-s + 265. i·29-s + (−8.85 + 5.11i)31-s + ⋯
L(s)  = 1  + (0.431 − 0.902i)3-s + (−0.518 + 0.898i)5-s + (0.993 + 0.112i)7-s + (−0.627 − 0.778i)9-s + (−0.425 + 0.245i)11-s + 1.33i·13-s + (0.587 + 0.855i)15-s + (0.152 + 0.264i)17-s + (−0.114 − 0.0662i)19-s + (0.530 − 0.847i)21-s + (0.542 + 0.313i)23-s + (−0.0385 − 0.0668i)25-s + (−0.972 + 0.230i)27-s + 1.70i·29-s + (−0.0513 + 0.0296i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.633 - 0.773i$
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: \((2,\ 336,\ (\ :3/2),\ 0.633 - 0.773i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.818243804\)
\(L(\frac12)\) \(\approx\) \(1.818243804\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.24 + 4.68i)T \)
7 \( 1 + (-18.4 - 2.09i)T \)
good5 \( 1 + (5.80 - 10.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (15.5 - 8.95i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 62.4iT - 2.19e3T^{2} \)
17 \( 1 + (-10.7 - 18.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (9.50 + 5.48i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-59.8 - 34.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 265. iT - 2.43e4T^{2} \)
31 \( 1 + (8.85 - 5.11i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (20.8 - 36.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 31.0T + 6.89e4T^{2} \)
43 \( 1 - 224.T + 7.95e4T^{2} \)
47 \( 1 + (-81.8 + 141. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-456. + 263. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-205. - 356. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-223. - 129. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-161. - 280. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 45.4iT - 3.57e5T^{2} \)
73 \( 1 + (486. - 281. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-144. + 250. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 448.T + 5.71e5T^{2} \)
89 \( 1 + (280. - 486. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 214. iT - 9.12e5T^{2} \)
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

−11.39169849941171699881778673519, −10.57820866832094693679580921607, −9.118526826256321124786868328875, −8.334255556945228407971353404111, −7.25748291227948730036468767682, −6.88233435036840257936812942644, −5.40117359294278861788251456664, −3.95481450751363973231452357901, −2.65836803083807922302193598516, −1.48642386162484006500971061297, 0.64578155283160375270690354003, 2.59422633061877871388284351459, 3.99387681509463886786834327904, 4.87552830426865712699904432389, 5.65269980970256431058294367630, 7.70106436085459786416878545793, 8.187436982924762638873549230995, 8.980274948836794367330897774889, 10.13795196281853396695106670031, 10.89427445827534719178146289544

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