↑

Properties

Label 2-336-84.23-c3-0-35
Degree $2$
Conductor $336$
Sign $0.440 + 0.897i$
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  + (2.89 + 4.31i)3-s + (−6.70 − 3.86i)5-s + (−13.9 − 12.2i)7-s + (−10.2 + 24.9i)9-s + (2.34 + 4.05i)11-s + 32.7·13-s + (−2.67 − 40.1i)15-s + (63.6 − 36.7i)17-s + (−79.1 − 45.7i)19-s + (12.5 − 95.4i)21-s + (70.4 − 122. i)23-s + (−32.5 − 56.4i)25-s + (−137. + 27.8i)27-s − 155. i·29-s + (76.0 − 43.9i)31-s + ⋯
L(s)  = 1  + (0.556 + 0.830i)3-s + (−0.599 − 0.346i)5-s + (−0.751 − 0.660i)7-s + (−0.380 + 0.924i)9-s + (0.0642 + 0.111i)11-s + 0.699·13-s + (−0.0461 − 0.690i)15-s + (0.907 − 0.523i)17-s + (−0.955 − 0.551i)19-s + (0.130 − 0.991i)21-s + (0.639 − 1.10i)23-s + (−0.260 − 0.451i)25-s + (−0.980 + 0.198i)27-s − 0.997i·29-s + (0.440 − 0.254i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.440 + 0.897i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.440 + 0.897i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.371986260\)
\(L(\frac12)\) \(\approx\) \(1.371986260\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.89 - 4.31i)T \)
7 \( 1 + (13.9 + 12.2i)T \)
good5 \( 1 + (6.70 + 3.86i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (-2.34 - 4.05i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 32.7T + 2.19e3T^{2} \)
17 \( 1 + (-63.6 + 36.7i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (79.1 + 45.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-70.4 + 122. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 155. iT - 2.43e4T^{2} \)
31 \( 1 + (-76.0 + 43.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-97.5 + 168. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 371. iT - 6.89e4T^{2} \)
43 \( 1 - 353. iT - 7.95e4T^{2} \)
47 \( 1 + (20.4 - 35.4i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-617. + 356. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (114. + 198. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (208. - 361. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (524. - 302. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 419.T + 3.57e5T^{2} \)
73 \( 1 + (82.5 + 143. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (486. + 280. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 447.T + 5.71e5T^{2} \)
89 \( 1 + (418. + 241. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.16e3T + 9.12e5T^{2} \)
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

−10.74899474890748838033526676794, −10.06282484834786510747090801789, −9.085428746778357622545573982910, −8.285374491838606349704295076792, −7.29469497196555065262420975246, −6.02519771769713667248484681040, −4.55361957512706541829916280427, −3.88702708056535951915688482983, −2.69297455421381060953833863400, −0.48013148368033625062384051656, 1.40783130089083474123125777903, 2.98875008133162386465731495931, 3.72725484965058646708833202312, 5.69104695777162535775386164450, 6.53318017964433505931439491841, 7.52463271993462012022222252672, 8.424089331991721966764013875992, 9.194816610822131389822332068133, 10.35458879047967200036307640790, 11.52749682737927827059379319780

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