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Properties

Label 2-6e3-72.29-c2-0-14
Degree $2$
Conductor $216$
Sign $0.964 - 0.263i$
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  + (1.95 − 0.423i)2-s + (3.64 − 1.65i)4-s + (3.64 + 6.31i)5-s + (−0.487 + 0.843i)7-s + (6.41 − 4.77i)8-s + (9.80 + 10.8i)10-s + (−7.19 + 12.4i)11-s + (5.50 − 3.17i)13-s + (−0.595 + 1.85i)14-s + (10.5 − 12.0i)16-s − 27.9i·17-s + 9.44i·19-s + (23.7 + 16.9i)20-s + (−8.78 + 27.4i)22-s + (−5.15 + 2.97i)23-s + ⋯
L(s)  = 1  + (0.977 − 0.211i)2-s + (0.910 − 0.413i)4-s + (0.729 + 1.26i)5-s + (−0.0695 + 0.120i)7-s + (0.802 − 0.596i)8-s + (0.980 + 1.08i)10-s + (−0.654 + 1.13i)11-s + (0.423 − 0.244i)13-s + (−0.0425 + 0.132i)14-s + (0.657 − 0.753i)16-s − 1.64i·17-s + 0.497i·19-s + (1.18 + 0.848i)20-s + (−0.399 + 1.24i)22-s + (−0.223 + 0.129i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.964 - 0.263i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.964 - 0.263i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.99759 + 0.401950i\)
\(L(\frac12)\) \(\approx\) \(2.99759 + 0.401950i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.95 + 0.423i)T \)
3 \( 1 \)
good5 \( 1 + (-3.64 - 6.31i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (0.487 - 0.843i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (7.19 - 12.4i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.50 + 3.17i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 27.9iT - 289T^{2} \)
19 \( 1 - 9.44iT - 361T^{2} \)
23 \( 1 + (5.15 - 2.97i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-24.9 + 43.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (11.0 + 19.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 20.4iT - 1.36e3T^{2} \)
41 \( 1 + (16.9 - 9.77i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (19.6 + 11.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (71.3 + 41.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 53.3T + 2.80e3T^{2} \)
59 \( 1 + (-29.5 - 51.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (9.09 + 5.24i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (44.8 - 25.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 11.6iT - 5.04e3T^{2} \)
73 \( 1 + 39.1T + 5.32e3T^{2} \)
79 \( 1 + (-49.7 + 86.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (3.61 - 6.26i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 97.4iT - 7.92e3T^{2} \)
97 \( 1 + (2.40 - 4.16i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.11528684237903093429134480019, −11.29875827474636920314660227007, −10.19734106906754090176848546647, −9.781945408766409824199978486224, −7.70360805516787476976138654784, −6.80214143060897356864657116669, −5.88316007658180310452634940717, −4.70568552433718195991468529920, −3.10398527552273747333509931360, −2.17286751856910725767169161524, 1.58740056428883517097223099339, 3.37982542831090005128926640973, 4.77249496353086005307103684479, 5.64729172387817024097264634622, 6.54349642870183805842409054614, 8.179022155515473931856279598622, 8.797995366462689228646309350553, 10.34664947102090247511618137709, 11.19124260262348317788761135072, 12.52027626585289566961286844032

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