↑

Properties

Label 2-336-112.109-c1-0-9
Degree $2$
Conductor $336$
Sign $0.912 + 0.408i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.453 − 1.33i)2-s + (0.965 − 0.258i)3-s + (−1.58 + 1.21i)4-s + (1.78 + 0.477i)5-s + (−0.784 − 1.17i)6-s + (0.786 + 2.52i)7-s + (2.34 + 1.57i)8-s + (0.866 − 0.499i)9-s + (−0.167 − 2.60i)10-s + (1.40 + 5.25i)11-s + (−1.22 + 1.58i)12-s + (−1.21 − 1.21i)13-s + (3.02 − 2.19i)14-s + 1.84·15-s + (1.05 − 3.85i)16-s + (−1.43 + 2.48i)17-s + ⋯
L(s)  = 1  + (−0.320 − 0.947i)2-s + (0.557 − 0.149i)3-s + (−0.794 + 0.606i)4-s + (0.797 + 0.213i)5-s + (−0.320 − 0.480i)6-s + (0.297 + 0.954i)7-s + (0.829 + 0.558i)8-s + (0.288 − 0.166i)9-s + (−0.0530 − 0.823i)10-s + (0.424 + 1.58i)11-s + (−0.352 + 0.457i)12-s + (−0.337 − 0.337i)13-s + (0.809 − 0.587i)14-s + 0.476·15-s + (0.263 − 0.964i)16-s + (−0.347 + 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.912 + 0.408i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.912 + 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42996 - 0.305354i\)
\(L(\frac12)\) \(\approx\) \(1.42996 - 0.305354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 + 1.33i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-0.786 - 2.52i)T \)
good5 \( 1 + (-1.78 - 0.477i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.40 - 5.25i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.21 + 1.21i)T + 13iT^{2} \)
17 \( 1 + (1.43 - 2.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.672 + 2.51i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.05 + 2.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.62 + 3.62i)T + 29iT^{2} \)
31 \( 1 + (-1.98 + 3.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.62 + 0.971i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.278iT - 41T^{2} \)
43 \( 1 + (-6.26 + 6.26i)T - 43iT^{2} \)
47 \( 1 + (3.06 + 5.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.620 - 2.31i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.55 - 5.80i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.03 + 15.0i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.81 - 1.82i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + (-4.57 - 2.63i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.89 + 5.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.5 - 10.5i)T + 83iT^{2} \)
89 \( 1 + (7.93 - 4.57i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.86T + 97T^{2} \)
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

−11.54583370811793241745808433956, −10.39568261958984546800911421699, −9.592102534883046637911798884312, −9.028312383291973421514200458993, −7.979314351090753459705404372086, −6.85400824564628832969764152608, −5.34538656850124764253493944180, −4.18797194806368154880187726864, −2.55363086761691831233824590806, −1.91586434283666756701641965169, 1.30393937114040568018611253837, 3.49407928779465210063094689146, 4.78997822321246672430372196991, 5.84150344195464917314339710545, 6.91632920844515753350073297173, 7.83383085031926424237086140246, 8.862139631933289867902187621406, 9.443885844507956087571872307507, 10.43360034627346679944383022718, 11.33033389235698286344650224325

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