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Properties

Label 2-2e5-8.5-c7-0-3
Degree $2$
Conductor $32$
Sign $0.857 + 0.513i$
Analytic cond. $9.99632$
Root an. cond. $3.16169$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 40.2i·3-s + 324. i·5-s + 956.·7-s + 569.·9-s − 5.45e3i·11-s − 6.28e3i·13-s + 1.30e4·15-s + 3.45e4·17-s + 1.45e4i·19-s − 3.84e4i·21-s + 2.46e4·23-s − 2.71e4·25-s − 1.10e5i·27-s + 1.71e5i·29-s − 1.11e5·31-s + ⋯
L(s)  = 1  − 0.859i·3-s + 1.16i·5-s + 1.05·7-s + 0.260·9-s − 1.23i·11-s − 0.793i·13-s + 0.998·15-s + 1.70·17-s + 0.488i·19-s − 0.906i·21-s + 0.422·23-s − 0.347·25-s − 1.08i·27-s + 1.30i·29-s − 0.673·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.513i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.857 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.857 + 0.513i$
Analytic conductor: \(9.99632\)
Root analytic conductor: \(3.16169\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :7/2),\ 0.857 + 0.513i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.94119 - 0.536973i\)
\(L(\frac12)\) \(\approx\) \(1.94119 - 0.536973i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 40.2iT - 2.18e3T^{2} \)
5 \( 1 - 324. iT - 7.81e4T^{2} \)
7 \( 1 - 956.T + 8.23e5T^{2} \)
11 \( 1 + 5.45e3iT - 1.94e7T^{2} \)
13 \( 1 + 6.28e3iT - 6.27e7T^{2} \)
17 \( 1 - 3.45e4T + 4.10e8T^{2} \)
19 \( 1 - 1.45e4iT - 8.93e8T^{2} \)
23 \( 1 - 2.46e4T + 3.40e9T^{2} \)
29 \( 1 - 1.71e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.11e5T + 2.75e10T^{2} \)
37 \( 1 + 1.03e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.16e4T + 1.94e11T^{2} \)
43 \( 1 - 3.28e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.19e5T + 5.06e11T^{2} \)
53 \( 1 + 1.04e6iT - 1.17e12T^{2} \)
59 \( 1 - 2.25e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.55e6iT - 3.14e12T^{2} \)
67 \( 1 + 3.16e5iT - 6.06e12T^{2} \)
71 \( 1 + 5.38e5T + 9.09e12T^{2} \)
73 \( 1 + 2.68e6T + 1.10e13T^{2} \)
79 \( 1 + 8.22e6T + 1.92e13T^{2} \)
83 \( 1 - 5.89e6iT - 2.71e13T^{2} \)
89 \( 1 - 4.37e5T + 4.42e13T^{2} \)
97 \( 1 + 7.84e6T + 8.07e13T^{2} \)
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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

−14.77216062182119151495401459518, −14.11774184109986775246027220360, −12.71417190014099770460378063826, −11.36647871761704734964409800463, −10.35064371880875854341313670993, −8.183953889847568431944729384834, −7.21701212440006962384602812542, −5.67346551674896208644101448274, −3.17791790466598936175492726342, −1.24965527033562442954500987866, 1.47233357333067625863462355272, 4.30330922035465681614105333630, 5.12002923692211082562771860120, 7.57580825931589858403040245539, 9.061706111576974386139262456565, 10.08137739312159876278443291319, 11.67262015276301913742300086757, 12.77714150327853309273502533641, 14.39203259543494557078906210496, 15.40519455865784082939914881480

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