↑

Properties

Label 2-336-28.23-c2-0-9
Degree $2$
Conductor $336$
Sign $-0.662 + 0.749i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (−2.29 + 3.98i)5-s + (−4.95 + 4.93i)7-s + (1.5 − 2.59i)9-s + (−0.161 + 0.0930i)11-s + 13.7·13-s − 7.96i·15-s + (−11.6 − 20.1i)17-s + (−28.5 − 16.4i)19-s + (3.16 − 11.7i)21-s + (−21.4 − 12.4i)23-s + (1.93 + 3.35i)25-s + 5.19i·27-s + 24.0·29-s + (−1.08 + 0.628i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (−0.459 + 0.796i)5-s + (−0.708 + 0.705i)7-s + (0.166 − 0.288i)9-s + (−0.0146 + 0.00846i)11-s + 1.05·13-s − 0.530i·15-s + (−0.685 − 1.18i)17-s + (−1.50 − 0.866i)19-s + (0.150 − 0.557i)21-s + (−0.934 − 0.539i)23-s + (0.0774 + 0.134i)25-s + 0.192i·27-s + 0.829·29-s + (−0.0351 + 0.0202i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ -0.662 + 0.749i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0216626 - 0.0480697i\)
\(L(\frac12)\) \(\approx\) \(0.0216626 - 0.0480697i\)
\(L(2)\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (4.95 - 4.93i)T \)
good5 \( 1 + (2.29 - 3.98i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (0.161 - 0.0930i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 13.7T + 169T^{2} \)
17 \( 1 + (11.6 + 20.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (28.5 + 16.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (21.4 + 12.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 24.0T + 841T^{2} \)
31 \( 1 + (1.08 - 0.628i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-17.7 + 30.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 77.2T + 1.68e3T^{2} \)
43 \( 1 - 41.9iT - 1.84e3T^{2} \)
47 \( 1 + (52.3 + 30.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (17.2 + 29.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-1.78 + 1.02i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (15.5 - 26.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-65.3 + 37.7i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 30.7iT - 5.04e3T^{2} \)
73 \( 1 + (-22.7 - 39.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (0.943 + 0.544i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 39.4iT - 6.88e3T^{2} \)
89 \( 1 + (76.1 - 131. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 25.9T + 9.40e3T^{2} \)
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

−11.09446189482885137417946433071, −10.22624626605414073010512394824, −9.164499164140963889514480765038, −8.282954221485162741673993585930, −6.71273390636618543430693573085, −6.41136587752478199735036368684, −4.97614621715309420963890014277, −3.72506609792636441053483513675, −2.53390569806915778612501926800, −0.02606434856725724641082274982, 1.54561915131039726440472573081, 3.70791905963022292277053633091, 4.50840313480849940043044990632, 6.05721014910654792724684183137, 6.62659155835684986420577977600, 8.101348574955213086417120025598, 8.590254237515451416471079943621, 10.06250789387830523124593620337, 10.70046726258293748309551857480, 11.75541631640869048948685693106

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