↑

Properties

Label 2-336-112.19-c1-0-11
Degree $2$
Conductor $336$
Sign $0.224 + 0.974i$
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  + (−0.904 − 1.08i)2-s + (−0.258 + 0.965i)3-s + (−0.361 + 1.96i)4-s + (−2.68 + 0.718i)5-s + (1.28 − 0.592i)6-s + (−0.164 − 2.64i)7-s + (2.46 − 1.38i)8-s + (−0.866 − 0.499i)9-s + (3.20 + 2.26i)10-s + (−0.476 + 1.77i)11-s + (−1.80 − 0.858i)12-s + (3.89 − 3.89i)13-s + (−2.72 + 2.56i)14-s − 2.77i·15-s + (−3.73 − 1.42i)16-s + (1.76 − 1.02i)17-s + ⋯
L(s)  = 1  + (−0.639 − 0.768i)2-s + (−0.149 + 0.557i)3-s + (−0.180 + 0.983i)4-s + (−1.19 + 0.321i)5-s + (0.524 − 0.242i)6-s + (−0.0622 − 0.998i)7-s + (0.871 − 0.490i)8-s + (−0.288 − 0.166i)9-s + (1.01 + 0.715i)10-s + (−0.143 + 0.536i)11-s + (−0.521 − 0.247i)12-s + (1.07 − 1.07i)13-s + (−0.727 + 0.686i)14-s − 0.716i·15-s + (−0.934 − 0.356i)16-s + (0.428 − 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.224 + 0.974i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.527354 - 0.419862i\)
\(L(\frac12)\) \(\approx\) \(0.527354 - 0.419862i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.904 + 1.08i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 + (0.164 + 2.64i)T \)
good5 \( 1 + (2.68 - 0.718i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.476 - 1.77i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.89 + 3.89i)T - 13iT^{2} \)
17 \( 1 + (-1.76 + 1.02i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.81 + 2.09i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.56 + 4.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.480 + 0.480i)T + 29iT^{2} \)
31 \( 1 + (3.39 + 5.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.55 - 9.53i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 9.53T + 41T^{2} \)
43 \( 1 + (3.82 + 3.82i)T + 43iT^{2} \)
47 \( 1 + (-6.70 + 11.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.39 + 1.17i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-8.96 - 2.40i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.502 + 1.87i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.26 - 0.606i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.35T + 71T^{2} \)
73 \( 1 + (-2.93 - 5.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.98 + 1.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.24 + 3.24i)T + 83iT^{2} \)
89 \( 1 + (-4.76 + 8.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.26iT - 97T^{2} \)
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

−11.34349426417140121265992926577, −10.40203246046648722011509026482, −9.864338335005272526385714120388, −8.541625298794794731331088051533, −7.72176783477227931425888618390, −6.97603943745679593848542978315, −5.01175066124430992518014003469, −3.79180273581318462618782381461, −3.19609407781152615271320633325, −0.70359511490517814266799326233, 1.35307577235409779322122011645, 3.54106969428242069945713855128, 5.17996996593108653146909982395, 6.00656657428105576382213939797, 7.17263179834100763014655188204, 7.976618547042290727238675174241, 8.738995029522912658739827835899, 9.490724912852177365252182476429, 11.09073835169476333587943569247, 11.57923456141366437214318085643

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