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Properties

Label 2-6e3-216.187-c2-0-47
Degree $2$
Conductor $216$
Sign $0.260 + 0.965i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.347 + 1.96i)2-s + (1.05 − 2.80i)3-s + (−3.75 − 1.36i)4-s + (5.16 + 3.04i)6-s + (4 − 6.92i)8-s + (−6.78 − 5.91i)9-s + (−5.63 − 4.72i)11-s + (−7.79 + 9.12i)12-s + (12.2 + 10.2i)16-s + (−16.8 − 29.2i)17-s + (13.9 − 11.3i)18-s + (7.56 − 13.1i)19-s + (11.2 − 9.45i)22-s + (−15.2 − 18.5i)24-s + (4.34 − 24.6i)25-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.350 − 0.936i)3-s + (−0.939 − 0.342i)4-s + (0.861 + 0.507i)6-s + (0.5 − 0.866i)8-s + (−0.754 − 0.656i)9-s + (−0.511 − 0.429i)11-s + (−0.649 + 0.760i)12-s + (0.766 + 0.642i)16-s + (−0.993 − 1.72i)17-s + (0.777 − 0.628i)18-s + (0.398 − 0.689i)19-s + (0.511 − 0.429i)22-s + (−0.635 − 0.771i)24-s + (0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.260 + 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.814040 - 0.623728i\)
\(L(\frac12)\) \(\approx\) \(0.814040 - 0.623728i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.347 - 1.96i)T \)
3 \( 1 + (-1.05 + 2.80i)T \)
good5 \( 1 + (-4.34 + 24.6i)T^{2} \)
7 \( 1 + (-37.5 + 31.4i)T^{2} \)
11 \( 1 + (5.63 + 4.72i)T + (21.0 + 119. i)T^{2} \)
13 \( 1 + (158. - 57.8i)T^{2} \)
17 \( 1 + (16.8 + 29.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-7.56 + 13.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-405. - 340. i)T^{2} \)
29 \( 1 + (790. + 287. i)T^{2} \)
31 \( 1 + (-736. - 617. i)T^{2} \)
37 \( 1 + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-1.45 - 8.26i)T + (-1.57e3 + 574. i)T^{2} \)
43 \( 1 + (-65.8 - 55.2i)T + (321. + 1.82e3i)T^{2} \)
47 \( 1 + (-1.69e3 + 1.41e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (89.9 - 75.4i)T + (604. - 3.42e3i)T^{2} \)
61 \( 1 + (-2.85e3 + 2.39e3i)T^{2} \)
67 \( 1 + (3.06 + 17.3i)T + (-4.21e3 + 1.53e3i)T^{2} \)
71 \( 1 + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (4.38 - 7.59i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (5.86e3 + 2.13e3i)T^{2} \)
83 \( 1 + (6.06 - 34.3i)T + (-6.47e3 - 2.35e3i)T^{2} \)
89 \( 1 + (-80.5 + 139. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-112. - 94.0i)T + (1.63e3 + 9.26e3i)T^{2} \)
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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

−12.09370549834611741304951185872, −10.93785593857411647121970844910, −9.466567065318784324629644280151, −8.741341316684824254730433223631, −7.69497665411416555192701748694, −6.95072788569363232282826130644, −5.92956186757591370277202385304, −4.66511324660676112703252278408, −2.76664632987421187016064879012, −0.57652199949209310578461029466, 2.07393500300647019319111274207, 3.50988510102735851226620486639, 4.46540518587400190944978266519, 5.68136452575949388448642307256, 7.70235049905091986650022072052, 8.679048373698843762589222509030, 9.489035285806213468116489413748, 10.52914197891063574863953276351, 10.93464148942839234030226880745, 12.22812102605961285647454383555

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