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Properties

Label 2-2e5-8.5-c17-0-7
Degree $2$
Conductor $32$
Sign $0.971 - 0.235i$
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  + 2.16e4i·3-s − 4.65e5i·5-s − 2.24e7·7-s − 3.40e8·9-s − 6.06e8i·11-s + 2.12e9i·13-s + 1.00e10·15-s − 5.45e9·17-s − 5.72e9i·19-s − 4.87e11i·21-s + 1.30e11·23-s + 5.46e11·25-s − 4.59e12i·27-s − 4.23e11i·29-s − 3.83e12·31-s + ⋯
L(s)  = 1  + 1.90i·3-s − 0.532i·5-s − 1.47·7-s − 2.64·9-s − 0.852i·11-s + 0.723i·13-s + 1.01·15-s − 0.189·17-s − 0.0773i·19-s − 2.81i·21-s + 0.346·23-s + 0.716·25-s − 3.13i·27-s − 0.157i·29-s − 0.808·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.971 - 0.235i$
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: \((2,\ 32,\ (\ :17/2),\ 0.971 - 0.235i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.9240023666\)
\(L(\frac12)\) \(\approx\) \(0.9240023666\)
\(L(\frac{19}{2})\) 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 \( 1 - 2.16e4iT - 1.29e8T^{2} \)
5 \( 1 + 4.65e5iT - 7.62e11T^{2} \)
7 \( 1 + 2.24e7T + 2.32e14T^{2} \)
11 \( 1 + 6.06e8iT - 5.05e17T^{2} \)
13 \( 1 - 2.12e9iT - 8.65e18T^{2} \)
17 \( 1 + 5.45e9T + 8.27e20T^{2} \)
19 \( 1 + 5.72e9iT - 5.48e21T^{2} \)
23 \( 1 - 1.30e11T + 1.41e23T^{2} \)
29 \( 1 + 4.23e11iT - 7.25e24T^{2} \)
31 \( 1 + 3.83e12T + 2.25e25T^{2} \)
37 \( 1 - 2.39e13iT - 4.56e26T^{2} \)
41 \( 1 + 5.79e12T + 2.61e27T^{2} \)
43 \( 1 - 4.00e13iT - 5.87e27T^{2} \)
47 \( 1 - 1.56e14T + 2.66e28T^{2} \)
53 \( 1 + 6.45e14iT - 2.05e29T^{2} \)
59 \( 1 + 8.37e14iT - 1.27e30T^{2} \)
61 \( 1 + 3.98e14iT - 2.24e30T^{2} \)
67 \( 1 - 5.22e15iT - 1.10e31T^{2} \)
71 \( 1 + 5.50e15T + 2.96e31T^{2} \)
73 \( 1 - 1.47e15T + 4.74e31T^{2} \)
79 \( 1 - 1.94e16T + 1.81e32T^{2} \)
83 \( 1 + 3.36e16iT - 4.21e32T^{2} \)
89 \( 1 + 1.09e16T + 1.37e33T^{2} \)
97 \( 1 - 6.57e16T + 5.95e33T^{2} \)
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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

−13.18817955337727723137340241037, −11.57691067126290029428786021180, −10.40618524484125505209292518165, −9.396459313823145449557475818743, −8.723584871470951305443468745168, −6.32024606368586445424870825937, −5.07718173747616932186338863285, −3.85850033676425879517937309638, −2.95747793056059024884860660180, −0.33508566193948355078414350778, 0.75171160588153527966549108123, 2.23064489265009335746794633203, 3.19480972058052020830246785278, 5.79513566853750361324760889571, 6.81745267785361884007861048457, 7.48489007080983332382621105832, 9.073836621681276871940414618877, 10.71223555474710960004482180882, 12.33000022749630752763491722848, 12.80591089514343716764329546948

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