Properties

Label 2-336-7.5-c6-0-35
Degree $2$
Conductor $336$
Sign $-0.980 - 0.195i$
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 + (−68.9 − 39.7i)5-s + (−284. − 191. i)7-s + (121.5 − 210. i)9-s + (411. + 712. i)11-s + 2.42e3i·13-s + 1.24e3·15-s + (6.75e3 − 3.89e3i)17-s + (−5.78e3 − 3.34e3i)19-s + (5.33e3 + 376. i)21-s + (9.41e3 − 1.63e4i)23-s + (−4.64e3 − 8.04e3i)25-s + 3.78e3i·27-s + 1.38e4·29-s + (2.41e4 − 1.39e4i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.551 − 0.318i)5-s + (−0.828 − 0.559i)7-s + (0.166 − 0.288i)9-s + (0.308 + 0.534i)11-s + 1.10i·13-s + 0.367·15-s + (1.37 − 0.793i)17-s + (−0.843 − 0.487i)19-s + (0.575 + 0.0406i)21-s + (0.773 − 1.34i)23-s + (−0.297 − 0.515i)25-s + 0.192i·27-s + 0.569·29-s + (0.810 − 0.467i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.980 - 0.195i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ -0.980 - 0.195i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.02555523831\)
\(L(\frac12)\) \(\approx\) \(0.02555523831\)
\(L(4)\) 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 \( 1 + (284. + 191. i)T \)
good5 \( 1 + (68.9 + 39.7i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-411. - 712. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 2.42e3iT - 4.82e6T^{2} \)
17 \( 1 + (-6.75e3 + 3.89e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (5.78e3 + 3.34e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-9.41e3 + 1.63e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 1.38e4T + 5.94e8T^{2} \)
31 \( 1 + (-2.41e4 + 1.39e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (3.98e4 - 6.90e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 5.91e4iT - 4.75e9T^{2} \)
43 \( 1 - 9.18e4T + 6.32e9T^{2} \)
47 \( 1 + (-4.34e3 - 2.50e3i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (9.31e4 + 1.61e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (1.95e5 - 1.12e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (1.25e5 + 7.21e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.17e5 + 2.03e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 9.62e4T + 1.28e11T^{2} \)
73 \( 1 + (-2.38e5 + 1.37e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (3.40e5 - 5.90e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 1.28e5iT - 3.26e11T^{2} \)
89 \( 1 + (3.22e5 + 1.86e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 6.20e5iT - 8.32e11T^{2} \)
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

−10.00401886024318021916011350319, −9.312319636390971076937064346516, −8.147843262451880322375634752250, −6.93116222007154496803549361610, −6.35999379402888048965411263915, −4.76958663127782957183811461413, −4.18903969489330793096847305243, −2.86451577753288930730843048332, −1.05896336442767658466613740802, −0.007927518594952216247566279742, 1.27255198060961256381470753052, 2.97750228926324295898233195216, 3.77455712049379599132322544173, 5.51218325693142719035304664403, 6.02640246939663150409744819207, 7.24168711220873688917414560353, 8.074381741401823512274928032052, 9.163392961282550129691554939370, 10.31024624033504238886804694053, 10.95278255812987882172073370186

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