↑

Properties

Label 2-336-7.3-c6-0-47
Degree $2$
Conductor $336$
Sign $-0.611 - 0.791i$
Analytic cond. $77.2981$
Root an. cond. $8.79193$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−13.5 − 7.79i)3-s + (113. − 65.4i)5-s + (−298. − 169. i)7-s + (121.5 + 210. i)9-s + (148. − 257. i)11-s − 1.92e3i·13-s − 2.03e3·15-s + (−573. − 331. i)17-s + (−5.17e3 + 2.99e3i)19-s + (2.70e3 + 4.60e3i)21-s + (−5.47e3 − 9.47e3i)23-s + (744. − 1.28e3i)25-s − 3.78e3i·27-s + 2.43e4·29-s + (−1.58e4 − 9.14e3i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (0.906 − 0.523i)5-s + (−0.869 − 0.493i)7-s + (0.166 + 0.288i)9-s + (0.111 − 0.193i)11-s − 0.874i·13-s − 0.604·15-s + (−0.116 − 0.0674i)17-s + (−0.755 + 0.435i)19-s + (0.292 + 0.497i)21-s + (−0.449 − 0.778i)23-s + (0.0476 − 0.0825i)25-s − 0.192i·27-s + 1.00·29-s + (−0.531 − 0.306i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.611 - 0.791i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ -0.611 - 0.791i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.1835660941\)
\(L(\frac12)\) \(\approx\) \(0.1835660941\)
\(L(4)\) 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 \( 1 + (13.5 + 7.79i)T \)
7 \( 1 + (298. + 169. i)T \)
good5 \( 1 + (-113. + 65.4i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-148. + 257. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 1.92e3iT - 4.82e6T^{2} \)
17 \( 1 + (573. + 331. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (5.17e3 - 2.99e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (5.47e3 + 9.47e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 2.43e4T + 5.94e8T^{2} \)
31 \( 1 + (1.58e4 + 9.14e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-6.44e3 - 1.11e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 6.79e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.20e5T + 6.32e9T^{2} \)
47 \( 1 + (-5.52e4 + 3.18e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (470. - 814. i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-2.05e5 - 1.18e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (5.76e4 - 3.32e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.02e3 + 1.77e3i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 6.35e5T + 1.28e11T^{2} \)
73 \( 1 + (5.74e5 + 3.31e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (9.29e4 + 1.60e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 2.98e5iT - 3.26e11T^{2} \)
89 \( 1 + (5.46e5 - 3.15e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.31e6iT - 8.32e11T^{2} \)
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

−10.14633897405177761710285494178, −9.081893787668091047951466801815, −8.055568158063439010332455186103, −6.78814694898408393084438743699, −6.05666347430072242408331871779, −5.16743634334897720940810281930, −3.84316381269460549152803503799, −2.40788040709556350662970369684, −1.08774148081665651024411130020, −0.04765513977844235000000666712, 1.74069838835064844463006974404, 2.84958621299390039535100245955, 4.17631603504390461368940184712, 5.44498789899780742163785480234, 6.37898379567926093240516397492, 6.87789261137104867797582843577, 8.549302962999286184438566783416, 9.578758388677913151264802088094, 10.02288367898859182791536605310, 11.07408393822840547704980697966

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