Properties

Label 2-3e3-27.25-c1-0-0
Degree $2$
Conductor $27$
Sign $0.381 - 0.924i$
Analytic cond. $0.215596$
Root an. cond. $0.464323$
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  + (−0.417 + 2.36i)2-s + (−0.210 − 1.71i)3-s + (−3.54 − 1.28i)4-s + (0.0713 − 0.0598i)5-s + (4.15 + 0.220i)6-s + (0.544 − 0.198i)7-s + (2.12 − 3.67i)8-s + (−2.91 + 0.722i)9-s + (0.111 + 0.193i)10-s + (−2.36 − 1.98i)11-s + (−1.47 + 6.35i)12-s + (0.729 + 4.13i)13-s + (0.241 + 1.37i)14-s + (−0.117 − 0.110i)15-s + (2.03 + 1.71i)16-s + (0.995 + 1.72i)17-s + ⋯
L(s)  = 1  + (−0.294 + 1.67i)2-s + (−0.121 − 0.992i)3-s + (−1.77 − 0.644i)4-s + (0.0318 − 0.0267i)5-s + (1.69 + 0.0898i)6-s + (0.205 − 0.0749i)7-s + (0.750 − 1.29i)8-s + (−0.970 + 0.240i)9-s + (0.0353 + 0.0612i)10-s + (−0.714 − 0.599i)11-s + (−0.424 + 1.83i)12-s + (0.202 + 1.14i)13-s + (0.0646 + 0.366i)14-s + (−0.0304 − 0.0284i)15-s + (0.509 + 0.427i)16-s + (0.241 + 0.418i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.381 - 0.924i$
Analytic conductor: \(0.215596\)
Root analytic conductor: \(0.464323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :1/2),\ 0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.456034 + 0.305088i\)
\(L(\frac12)\) \(\approx\) \(0.456034 + 0.305088i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.210 + 1.71i)T \)
good2 \( 1 + (0.417 - 2.36i)T + (-1.87 - 0.684i)T^{2} \)
5 \( 1 + (-0.0713 + 0.0598i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.544 + 0.198i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.36 + 1.98i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.729 - 4.13i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.995 - 1.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.92 + 3.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.18 - 1.52i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.11 + 6.30i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.55 + 0.566i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (2.01 + 3.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.190 + 1.07i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (5.28 + 4.43i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (3.37 - 1.22i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 5.40T + 53T^{2} \)
59 \( 1 + (7.87 - 6.61i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-12.4 + 4.51i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.53 + 8.70i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (0.572 + 0.991i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.0977 - 0.169i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.25 - 7.09i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (2.58 - 14.6i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-0.776 + 1.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.05 - 3.40i)T + (16.8 + 95.5i)T^{2} \)
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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

−17.42399262472809085136300549003, −16.61819122944113179353998044525, −15.36041552033424918467201891674, −14.05404275967019919056874332818, −13.25779482656238762043850678158, −11.37031216284053679685330909468, −9.072696096482557683720736346120, −7.86813441557953582351069766669, −6.76068675182425559962395746934, −5.40795628570150497123888584382, 3.14180862711550723881417911933, 4.99438736298409655793202273734, 8.428025627091345929483898172232, 9.910723521016106146839225707299, 10.57421682883284338796477237228, 11.81656664049646968336630172615, 13.01500691952281668963182732381, 14.64897852016930633147744427346, 16.07046329049329325618383054303, 17.64516651764500627543085725488

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