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Properties

Label 2-6e3-216.187-c0-0-0
Degree $2$
Conductor $216$
Sign $0.727 + 0.686i$
Analytic cond. $0.107798$
Root an. cond. $0.328326$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−1.43 − 1.20i)11-s + (−0.499 − 0.866i)12-s + (0.766 + 0.642i)16-s + (−0.173 − 0.300i)17-s + 18-s + (−0.766 + 1.32i)19-s + (−1.43 + 1.20i)22-s + (−0.939 + 0.342i)24-s + (0.173 − 0.984i)25-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−1.43 − 1.20i)11-s + (−0.499 − 0.866i)12-s + (0.766 + 0.642i)16-s + (−0.173 − 0.300i)17-s + 18-s + (−0.766 + 1.32i)19-s + (−1.43 + 1.20i)22-s + (−0.939 + 0.342i)24-s + (0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(0.107798\)
Root analytic conductor: \(0.328326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :0),\ 0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8249248585\)
\(L(\frac12)\) \(\approx\) \(0.8249248585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
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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

−12.55124754139558468556378465834, −11.21051668293725209554342621344, −10.50203028351965909091085822696, −9.783166078178654752475383996015, −8.527100627928353288891685889834, −8.036009564540510546769230192679, −5.84167590691965716889083289510, −4.70295673344309367410168992789, −3.47532539152169408503793629324, −2.41271315573078278806371546827, 2.56436732427807162727599322871, 4.22934562509257204306838507095, 5.48389329429040903075294870649, 6.93695020489685553561509634112, 7.45879046930968058463061176450, 8.522728591203539970167515359401, 9.351766908486710837419905923422, 10.52523905869588425920620339523, 12.21643200652767165625266610959, 13.05948287739130373666504063540

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