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Properties

Label 2-3e2-9.5-c4-0-2
Degree $2$
Conductor $9$
Sign $0.936 + 0.350i$
Analytic cond. $0.930329$
Root an. cond. $0.964535$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.86 − 2.23i)2-s + (−8.70 + 2.29i)3-s + (1.96 − 3.41i)4-s + (13.8 + 8.01i)5-s + (−28.5 + 28.2i)6-s + (−36.2 − 62.7i)7-s + 53.8i·8-s + (70.4 − 39.8i)9-s + 71.5·10-s + (83.2 − 48.0i)11-s + (−9.32 + 34.1i)12-s + (−76.9 + 133. i)13-s + (−280. − 161. i)14-s + (−139. − 37.9i)15-s + (151. + 262. i)16-s + 72.7i·17-s + ⋯
L(s)  = 1  + (0.966 − 0.558i)2-s + (−0.967 + 0.254i)3-s + (0.123 − 0.213i)4-s + (0.555 + 0.320i)5-s + (−0.792 + 0.785i)6-s + (−0.739 − 1.28i)7-s + 0.841i·8-s + (0.870 − 0.492i)9-s + 0.715·10-s + (0.688 − 0.397i)11-s + (−0.0647 + 0.237i)12-s + (−0.455 + 0.788i)13-s + (−1.43 − 0.825i)14-s + (−0.618 − 0.168i)15-s + (0.592 + 1.02i)16-s + 0.251i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(0.930329\)
Root analytic conductor: \(0.964535\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :2),\ 0.936 + 0.350i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.18134 - 0.213587i\)
\(L(\frac12)\) \(\approx\) \(1.18134 - 0.213587i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (8.70 - 2.29i)T \)
good2 \( 1 + (-3.86 + 2.23i)T + (8 - 13.8i)T^{2} \)
5 \( 1 + (-13.8 - 8.01i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (36.2 + 62.7i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-83.2 + 48.0i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (76.9 - 133. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 72.7iT - 8.35e4T^{2} \)
19 \( 1 + 190.T + 1.30e5T^{2} \)
23 \( 1 + (12.5 + 7.22i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-620. + 358. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-151. + 262. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 826.T + 1.87e6T^{2} \)
41 \( 1 + (-481. - 278. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (446. + 773. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (3.42e3 - 1.97e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 1.96e3iT - 7.89e6T^{2} \)
59 \( 1 + (-4.68e3 - 2.70e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-856. - 1.48e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-2.31e3 + 4.01e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 6.69e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.82e3T + 2.83e7T^{2} \)
79 \( 1 + (-2.86e3 - 4.96e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-2.45e3 + 1.41e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 1.42e4iT - 6.27e7T^{2} \)
97 \( 1 + (3.58e3 + 6.20e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−21.17257045061050733430270798448, −19.55921635606865646340862821730, −17.49049739679357237804995811983, −16.56037340269829390602535055783, −14.27773302045504918850079722709, −13.04240561798579688180729478161, −11.52581995196533337180556170483, −10.08940977307677130487187661268, −6.43733661612261843547882195491, −4.16441993780772380695722002942, 5.25087573306371985183922040991, 6.46389251052961450853473964807, 9.721693755352397029136819496706, 12.20779491062379161780508694691, 13.11535760564479381417601507351, 15.01511657662431838051204377807, 16.25446873641872907717733855138, 17.78354800038124479151500074824, 19.20673734716847297286548919708, 21.60026041560796255720634589346

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