Properties

Label 2-6e3-216.13-c1-0-17
Degree $2$
Conductor $216$
Sign $0.0954 + 0.995i$
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  + (−1.39 − 0.213i)2-s + (−0.479 − 1.66i)3-s + (1.90 + 0.596i)4-s + (2.25 − 2.69i)5-s + (0.314 + 2.42i)6-s + (3.36 + 1.22i)7-s + (−2.54 − 1.24i)8-s + (−2.54 + 1.59i)9-s + (−3.73 + 3.27i)10-s + (1.86 + 2.21i)11-s + (0.0783 − 3.46i)12-s + (−1.53 − 0.270i)13-s + (−4.44 − 2.43i)14-s + (−5.56 − 2.46i)15-s + (3.28 + 2.27i)16-s + (3.86 − 6.69i)17-s + ⋯
L(s)  = 1  + (−0.988 − 0.150i)2-s + (−0.276 − 0.960i)3-s + (0.954 + 0.298i)4-s + (1.00 − 1.20i)5-s + (0.128 + 0.991i)6-s + (1.27 + 0.463i)7-s + (−0.898 − 0.438i)8-s + (−0.846 + 0.531i)9-s + (−1.17 + 1.03i)10-s + (0.561 + 0.669i)11-s + (0.0226 − 0.999i)12-s + (−0.425 − 0.0750i)13-s + (−1.18 − 0.650i)14-s + (−1.43 − 0.637i)15-s + (0.822 + 0.569i)16-s + (0.937 − 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0954 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0954 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.0954 + 0.995i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.0954 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693091 - 0.629792i\)
\(L(\frac12)\) \(\approx\) \(0.693091 - 0.629792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.213i)T \)
3 \( 1 + (0.479 + 1.66i)T \)
good5 \( 1 + (-2.25 + 2.69i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-3.36 - 1.22i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.86 - 2.21i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (1.53 + 0.270i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-3.86 + 6.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.847 - 0.489i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.90 - 2.51i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (5.11 - 0.902i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.11 - 0.770i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.179 - 0.103i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.00 - 5.68i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-2.62 - 3.12i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-1.96 - 0.715i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 3.19iT - 53T^{2} \)
59 \( 1 + (2.41 - 2.88i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.94 + 8.08i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.94 - 0.519i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (4.82 - 8.35i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.70 - 2.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.447 + 2.53i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-6.28 + 1.10i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-4.56 - 7.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.10 + 5.96i)T + (16.8 - 95.5i)T^{2} \)
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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

−12.01950219169927424981733747250, −11.37038957053458848279418207405, −9.848147345013608158925747878011, −9.143560594753753361848657035836, −8.112560998067081354997236104823, −7.35913104307882994323270946916, −5.90516467154481621549234328670, −5.04608835892832027164495847012, −2.18919715726850946089899880234, −1.32191283210873243407365367368, 1.99892054127417667242061212688, 3.73271357498884768236603255853, 5.61746342830268872671648349153, 6.30872618875404299215027882622, 7.68540258138067730033688045620, 8.730280829134763385976665260549, 9.888390550851801026445392548946, 10.53111350993760583563514642690, 11.02546947360397732552112565960, 12.01670887928480139887563734877

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