Properties

Label 2-2e5-8.5-c7-0-5
Degree $2$
Conductor $32$
Sign $-0.985 + 0.167i$
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  − 76.9i·3-s − 338. i·5-s + 438.·7-s − 3.73e3·9-s + 1.96e3i·11-s + 2.21e3i·13-s − 2.60e4·15-s − 1.21e4·17-s − 3.28e4i·19-s − 3.37e4i·21-s − 1.96e4·23-s − 3.64e4·25-s + 1.19e5i·27-s + 1.60e5i·29-s + 2.29e5·31-s + ⋯
L(s)  = 1  − 1.64i·3-s − 1.21i·5-s + 0.483·7-s − 1.70·9-s + 0.445i·11-s + 0.279i·13-s − 1.99·15-s − 0.598·17-s − 1.09i·19-s − 0.795i·21-s − 0.335·23-s − 0.466·25-s + 1.16i·27-s + 1.22i·29-s + 1.38·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.985 + 0.167i$
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: \((2,\ 32,\ (\ :7/2),\ -0.985 + 0.167i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.122934 - 1.45514i\)
\(L(\frac12)\) \(\approx\) \(0.122934 - 1.45514i\)
\(L(\frac{9}{2})\) 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 \( 1 + 76.9iT - 2.18e3T^{2} \)
5 \( 1 + 338. iT - 7.81e4T^{2} \)
7 \( 1 - 438.T + 8.23e5T^{2} \)
11 \( 1 - 1.96e3iT - 1.94e7T^{2} \)
13 \( 1 - 2.21e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.21e4T + 4.10e8T^{2} \)
19 \( 1 + 3.28e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.96e4T + 3.40e9T^{2} \)
29 \( 1 - 1.60e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.29e5T + 2.75e10T^{2} \)
37 \( 1 + 4.96e5iT - 9.49e10T^{2} \)
41 \( 1 - 5.99e5T + 1.94e11T^{2} \)
43 \( 1 + 8.83e4iT - 2.71e11T^{2} \)
47 \( 1 + 8.20e5T + 5.06e11T^{2} \)
53 \( 1 + 1.53e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.82e6iT - 2.48e12T^{2} \)
61 \( 1 + 4.84e5iT - 3.14e12T^{2} \)
67 \( 1 - 7.98e4iT - 6.06e12T^{2} \)
71 \( 1 + 1.27e6T + 9.09e12T^{2} \)
73 \( 1 - 3.70e6T + 1.10e13T^{2} \)
79 \( 1 - 2.55e6T + 1.92e13T^{2} \)
83 \( 1 + 1.53e6iT - 2.71e13T^{2} \)
89 \( 1 - 1.99e6T + 4.42e13T^{2} \)
97 \( 1 + 2.89e4T + 8.07e13T^{2} \)
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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

−14.33209996447555037023327097422, −13.15386798910783165390003917362, −12.49559070353605469816910561760, −11.36119240918799067039979003302, −9.049433442789260541264163078142, −7.962540343840501559336118560231, −6.65888563982908288301448129517, −4.91564704537888073003004333394, −2.00927200636710120099327563999, −0.69962857083990405606003934653, 2.98366391105781296884562579858, 4.39242488855139775717314783538, 6.09447505341967614494552090934, 8.169044837161255900095474211360, 9.801030355021572576741838527541, 10.64804278403974038528350927019, 11.58128283502675117952697091029, 13.90265782932642654623163269443, 14.86522233846568473604788621652, 15.59599693249637905423784086291

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