↑

Properties

Label 2-336-7.5-c6-0-19
Degree $2$
Conductor $336$
Sign $0.985 - 0.167i$
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 + (−65.9 − 38.1i)5-s + (334. + 77.8i)7-s + (121.5 − 210. i)9-s + (361. + 626. i)11-s − 681. i·13-s + 1.18e3·15-s + (−645. + 372. i)17-s + (3.27e3 + 1.89e3i)19-s + (−5.11e3 + 1.55e3i)21-s + (1.73e3 − 3.00e3i)23-s + (−4.90e3 − 8.50e3i)25-s + 3.78e3i·27-s + 1.62e4·29-s + (−3.98e4 + 2.30e4i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.527 − 0.304i)5-s + (0.973 + 0.226i)7-s + (0.166 − 0.288i)9-s + (0.271 + 0.470i)11-s − 0.310i·13-s + 0.351·15-s + (−0.131 + 0.0758i)17-s + (0.477 + 0.275i)19-s + (−0.552 + 0.167i)21-s + (0.142 − 0.246i)23-s + (−0.314 − 0.544i)25-s + 0.192i·27-s + 0.664·29-s + (−1.33 + 0.772i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.985 - 0.167i$
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: \((2,\ 336,\ (\ :3),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.707083022\)
\(L(\frac12)\) \(\approx\) \(1.707083022\)
\(L(4)\) 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 \( 1 + (-334. - 77.8i)T \)
good5 \( 1 + (65.9 + 38.1i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-361. - 626. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 681. iT - 4.82e6T^{2} \)
17 \( 1 + (645. - 372. i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-3.27e3 - 1.89e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-1.73e3 + 3.00e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 1.62e4T + 5.94e8T^{2} \)
31 \( 1 + (3.98e4 - 2.30e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-2.99e4 + 5.18e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 2.22e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.37e5T + 6.32e9T^{2} \)
47 \( 1 + (-1.74e4 - 1.00e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (7.46e4 + 1.29e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-2.38e5 + 1.37e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (2.45e4 + 1.41e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.83e5 - 3.17e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 6.27e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.62e4 + 2.09e4i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (1.70e5 - 2.95e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 9.97e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.11e6 - 6.41e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 1.34e6iT - 8.32e11T^{2} \)
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

−10.68722512163131818914704659229, −9.695237897695904988488672366060, −8.603916349018493663729851657843, −7.81431551317012780803020191257, −6.71992472631678684858685588595, −5.42889361603928488670184554107, −4.68878045531864925260570331390, −3.64649101105955930103242814574, −1.96977008273411030581430529070, −0.69364944797848598183906588812, 0.69940839274835253977069717131, 1.86181309665869193692649625954, 3.43230258321002983905118604106, 4.58365895387008377870093935232, 5.57917566684455103132455072028, 6.78337501412157901906980705512, 7.59371596564306474110764052183, 8.453364996648273988310787502782, 9.627753565113711969428808867688, 10.85845823976100927551195460727

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