Properties

Label 2-6e3-72.61-c1-0-6
Degree $2$
Conductor $216$
Sign $0.911 - 0.410i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 0.436i)2-s + (1.61 + 1.17i)4-s + (0.602 − 0.348i)5-s + (0.795 − 1.37i)7-s + (1.66 + 2.28i)8-s + (0.962 − 0.205i)10-s + (−2.37 − 1.36i)11-s + (−4.76 + 2.75i)13-s + (1.67 − 1.50i)14-s + (1.24 + 3.80i)16-s + 5.65·17-s + 0.963i·19-s + (1.38 + 0.143i)20-s + (−2.59 − 2.87i)22-s + (−3.28 − 5.69i)23-s + ⋯
L(s)  = 1  + (0.951 + 0.308i)2-s + (0.809 + 0.586i)4-s + (0.269 − 0.155i)5-s + (0.300 − 0.520i)7-s + (0.589 + 0.807i)8-s + (0.304 − 0.0649i)10-s + (−0.715 − 0.412i)11-s + (−1.32 + 0.763i)13-s + (0.446 − 0.402i)14-s + (0.311 + 0.950i)16-s + 1.37·17-s + 0.221i·19-s + (0.309 + 0.0320i)20-s + (−0.553 − 0.613i)22-s + (−0.685 − 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.911 - 0.410i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03804 + 0.437715i\)
\(L(\frac12)\) \(\approx\) \(2.03804 + 0.437715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.436i)T \)
3 \( 1 \)
good5 \( 1 + (-0.602 + 0.348i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.795 + 1.37i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.37 + 1.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.76 - 2.75i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 - 0.963iT - 19T^{2} \)
23 \( 1 + (3.28 + 5.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.85 + 1.64i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.69 + 6.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.25iT - 37T^{2} \)
41 \( 1 + (-0.931 - 1.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.99 + 1.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.85 - 6.67i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.54iT - 53T^{2} \)
59 \( 1 + (-4.62 + 2.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.93 - 4.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.95 + 3.43i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 - 2.83T + 73T^{2} \)
79 \( 1 + (-2.87 + 4.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.74 - 3.31i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.98T + 89T^{2} \)
97 \( 1 + (1.24 - 2.16i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.51746965978469741749521010571, −11.65824582684075570319972053826, −10.62210130171800166256965759590, −9.566375580381083463088899564811, −7.962631586085688718878546808394, −7.35516411268289458904846833851, −5.97341151031612049289243114809, −5.03447353466802988600281321278, −3.85710264371757370448366409336, −2.27448570432677294920759657740, 2.10449645220910708925499778287, 3.34563926440570693115614320553, 5.07003763217336351687393123120, 5.57399443997063628978831329702, 7.08460542583793968935398452423, 8.018209389671570424339626447663, 9.837911526446186697855237083228, 10.24464376480841083442529579686, 11.60903862029326554269371315758, 12.29140411049792877142683037013

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