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Properties

Label 2-2e5-8.5-c17-0-14
Degree $2$
Conductor $32$
Sign $-0.553 + 0.832i$
Analytic cond. $58.6310$
Root an. cond. $7.65709$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.34e4i·3-s − 1.59e6i·5-s + 1.66e7·7-s − 5.26e7·9-s − 6.79e8i·11-s − 3.13e9i·13-s + 2.14e10·15-s − 1.26e10·17-s − 5.87e9i·19-s + 2.23e11i·21-s − 5.43e11·23-s − 1.77e12·25-s + 1.03e12i·27-s + 2.82e12i·29-s − 8.42e11·31-s + ⋯
L(s)  = 1  + 1.18i·3-s − 1.82i·5-s + 1.08·7-s − 0.407·9-s − 0.955i·11-s − 1.06i·13-s + 2.16·15-s − 0.440·17-s − 0.0794i·19-s + 1.29i·21-s − 1.44·23-s − 2.32·25-s + 0.702i·27-s + 1.04i·29-s − 0.177·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(58.6310\)
Root analytic conductor: \(7.65709\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :17/2),\ -0.553 + 0.832i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.329902714\)
\(L(\frac12)\) \(\approx\) \(1.329902714\)
\(L(\frac{19}{2})\) 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 \( 1 - 1.34e4iT - 1.29e8T^{2} \)
5 \( 1 + 1.59e6iT - 7.62e11T^{2} \)
7 \( 1 - 1.66e7T + 2.32e14T^{2} \)
11 \( 1 + 6.79e8iT - 5.05e17T^{2} \)
13 \( 1 + 3.13e9iT - 8.65e18T^{2} \)
17 \( 1 + 1.26e10T + 8.27e20T^{2} \)
19 \( 1 + 5.87e9iT - 5.48e21T^{2} \)
23 \( 1 + 5.43e11T + 1.41e23T^{2} \)
29 \( 1 - 2.82e12iT - 7.25e24T^{2} \)
31 \( 1 + 8.42e11T + 2.25e25T^{2} \)
37 \( 1 - 6.17e12iT - 4.56e26T^{2} \)
41 \( 1 + 6.38e13T + 2.61e27T^{2} \)
43 \( 1 + 5.41e13iT - 5.87e27T^{2} \)
47 \( 1 - 3.89e13T + 2.66e28T^{2} \)
53 \( 1 + 5.92e14iT - 2.05e29T^{2} \)
59 \( 1 - 7.11e14iT - 1.27e30T^{2} \)
61 \( 1 + 9.01e14iT - 2.24e30T^{2} \)
67 \( 1 + 8.96e14iT - 1.10e31T^{2} \)
71 \( 1 + 8.49e13T + 2.96e31T^{2} \)
73 \( 1 - 1.01e16T + 4.74e31T^{2} \)
79 \( 1 + 3.99e15T + 1.81e32T^{2} \)
83 \( 1 - 2.19e15iT - 4.21e32T^{2} \)
89 \( 1 + 6.74e16T + 1.37e33T^{2} \)
97 \( 1 + 1.67e15T + 5.95e33T^{2} \)
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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

−12.59084151188838420633994166556, −11.31792898793396721984485990987, −10.07919617422476435994112121957, −8.787009957424773448592277690855, −8.142808121083856448053064173970, −5.48799391710518248709907992851, −4.84043905396959466744169257232, −3.73712232775375569119769860075, −1.56826747904660856578488234707, −0.31759163005869411319368519801, 1.74367204341610651853069377865, 2.32127842374366158279856044279, 4.19278313458458889606886463670, 6.26578812038659010045177018565, 7.11944892772636516158886990960, 7.933751299241260682078052089756, 9.963665077249629419314361890011, 11.25645893292451485834011779340, 12.06020272851260216132967749705, 13.71966263717017751380675122153

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