Properties

Label 2-6e3-216.85-c1-0-29
Degree $2$
Conductor $216$
Sign $-0.460 + 0.887i$
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  + (0.531 − 1.31i)2-s + (0.346 − 1.69i)3-s + (−1.43 − 1.39i)4-s + (3.12 + 0.550i)5-s + (−2.04 − 1.35i)6-s + (0.272 − 0.228i)7-s + (−2.58 + 1.14i)8-s + (−2.76 − 1.17i)9-s + (2.37 − 3.79i)10-s + (0.752 − 0.132i)11-s + (−2.86 + 1.95i)12-s + (1.01 + 2.79i)13-s + (−0.154 − 0.478i)14-s + (2.01 − 5.10i)15-s + (0.121 + 3.99i)16-s + (−3.30 + 5.72i)17-s + ⋯
L(s)  = 1  + (0.375 − 0.926i)2-s + (0.199 − 0.979i)3-s + (−0.717 − 0.696i)4-s + (1.39 + 0.246i)5-s + (−0.833 − 0.553i)6-s + (0.102 − 0.0863i)7-s + (−0.914 + 0.403i)8-s + (−0.920 − 0.391i)9-s + (0.752 − 1.20i)10-s + (0.226 − 0.0399i)11-s + (−0.825 + 0.564i)12-s + (0.282 + 0.775i)13-s + (−0.0413 − 0.127i)14-s + (0.519 − 1.31i)15-s + (0.0304 + 0.999i)16-s + (−0.802 + 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.460 + 0.887i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863272 - 1.42064i\)
\(L(\frac12)\) \(\approx\) \(0.863272 - 1.42064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.531 + 1.31i)T \)
3 \( 1 + (-0.346 + 1.69i)T \)
good5 \( 1 + (-3.12 - 0.550i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.272 + 0.228i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.752 + 0.132i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-1.01 - 2.79i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.30 - 5.72i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.123 - 0.0714i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.43 + 4.55i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.23 + 6.14i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.36 - 1.98i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-7.84 - 4.52i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.14 - 1.14i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-7.77 + 1.37i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.24 + 4.40i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 2.40iT - 53T^{2} \)
59 \( 1 + (10.8 + 1.90i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.68 + 5.58i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.33 + 9.16i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (6.07 - 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.95 + 3.39i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.515 - 0.187i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (4.63 - 12.7i)T + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (6.70 + 11.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.03 - 11.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−12.17770978991108537267189751959, −11.12408710458225172630384125139, −10.18605888969698189077790100141, −9.198699456203980865416878337584, −8.260286263883385195053798311171, −6.33065597896725432643630568873, −6.08655088920676536398327560974, −4.28522988028813271952648668856, −2.52735501141524712740262414210, −1.63190547492844215091295127998, 2.80509519424494543861848968671, 4.37957519969854728857180152379, 5.42787326908000212096336739949, 6.10285172681240005197504110097, 7.62193429247847411467965659780, 8.935424348378793078312611978755, 9.407508325617893341494563381871, 10.39941550027123899145562654656, 11.73735741514905748048219688047, 13.06689317348352239768809821208

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