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Properties

Label 2-3e2-9.5-c4-0-0
Degree $2$
Conductor $9$
Sign $-0.434 - 0.900i$
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  + (−5.67 + 3.27i)2-s + (−1.11 + 8.93i)3-s + (13.4 − 23.3i)4-s + (10.2 + 5.89i)5-s + (−22.9 − 54.3i)6-s + (26.6 + 46.1i)7-s + 71.6i·8-s + (−78.4 − 19.9i)9-s − 77.2·10-s + (108. − 62.4i)11-s + (193. + 146. i)12-s + (37.3 − 64.6i)13-s + (−302. − 174. i)14-s + (−64.0 + 84.5i)15-s + (−19.2 − 33.3i)16-s − 7.70i·17-s + ⋯
L(s)  = 1  + (−1.41 + 0.819i)2-s + (−0.124 + 0.992i)3-s + (0.841 − 1.45i)4-s + (0.408 + 0.235i)5-s + (−0.636 − 1.50i)6-s + (0.543 + 0.941i)7-s + 1.11i·8-s + (−0.969 − 0.246i)9-s − 0.772·10-s + (0.894 − 0.516i)11-s + (1.34 + 1.01i)12-s + (0.220 − 0.382i)13-s + (−1.54 − 0.890i)14-s + (−0.284 + 0.375i)15-s + (−0.0752 − 0.130i)16-s − 0.0266i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $-0.434 - 0.900i$
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: \((2,\ 9,\ (\ :2),\ -0.434 - 0.900i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.300688 + 0.478629i\)
\(L(\frac12)\) \(\approx\) \(0.300688 + 0.478629i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.11 - 8.93i)T \)
good2 \( 1 + (5.67 - 3.27i)T + (8 - 13.8i)T^{2} \)
5 \( 1 + (-10.2 - 5.89i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (-26.6 - 46.1i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-108. + 62.4i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-37.3 + 64.6i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 7.70iT - 8.35e4T^{2} \)
19 \( 1 + 54.1T + 1.30e5T^{2} \)
23 \( 1 + (-346. - 199. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (468. - 270. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-766. + 1.32e3i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.71e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.09e3 - 629. i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.30e3 - 2.25e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (692. - 400. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 4.22e3iT - 7.89e6T^{2} \)
59 \( 1 + (2.88e3 + 1.66e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-7.50 - 13.0i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (2.59e3 - 4.48e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 1.92e3iT - 2.54e7T^{2} \)
73 \( 1 - 949.T + 2.83e7T^{2} \)
79 \( 1 + (-118. - 206. i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (1.14e4 - 6.58e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 575. iT - 6.27e7T^{2} \)
97 \( 1 + (7.56e3 + 1.30e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−21.12533244589575764919901964011, −19.39436813152561489970413526472, −17.98678855297121965151283662077, −16.98568305644517455714220455584, −15.68740555355982170701183770750, −14.61090413774006195410523688790, −11.24982781826406031449774784490, −9.657344127998523603231432929950, −8.497175320800506517831948013721, −5.97151479953988768215279134759, 1.43416954902843804324915323752, 7.25259808378858088326653754619, 8.935528831918611667198271517288, 10.77997602641107249984782359224, 12.15948300402956413529349975776, 13.96043669200756142979617557366, 17.04917694968387109330175563809, 17.48118799453427654505527479428, 18.89344862444938517608262025846, 19.93516897880718329856908314765

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