Properties

Label 2-2e5-8.5-c17-0-6
Degree $2$
Conductor $32$
Sign $0.907 - 0.419i$
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  − 4.24e3i·3-s − 6.63e5i·5-s + 1.66e7·7-s + 1.11e8·9-s + 1.05e9i·11-s + 9.19e7i·13-s − 2.82e9·15-s − 1.98e10·17-s + 8.44e10i·19-s − 7.08e10i·21-s + 2.72e10·23-s + 3.22e11·25-s − 1.02e12i·27-s + 3.75e12i·29-s − 5.36e12·31-s + ⋯
L(s)  = 1  − 0.373i·3-s − 0.760i·5-s + 1.09·7-s + 0.860·9-s + 1.48i·11-s + 0.0312i·13-s − 0.284·15-s − 0.689·17-s + 1.14i·19-s − 0.408i·21-s + 0.0724·23-s + 0.422·25-s − 0.695i·27-s + 1.39i·29-s − 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.907 - 0.419i$
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: \((2,\ 32,\ (\ :17/2),\ 0.907 - 0.419i)\)

Particular Values

\(L(9)\) \(\approx\) \(2.445473756\)
\(L(\frac12)\) \(\approx\) \(2.445473756\)
\(L(\frac{19}{2})\) 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 \( 1 + 4.24e3iT - 1.29e8T^{2} \)
5 \( 1 + 6.63e5iT - 7.62e11T^{2} \)
7 \( 1 - 1.66e7T + 2.32e14T^{2} \)
11 \( 1 - 1.05e9iT - 5.05e17T^{2} \)
13 \( 1 - 9.19e7iT - 8.65e18T^{2} \)
17 \( 1 + 1.98e10T + 8.27e20T^{2} \)
19 \( 1 - 8.44e10iT - 5.48e21T^{2} \)
23 \( 1 - 2.72e10T + 1.41e23T^{2} \)
29 \( 1 - 3.75e12iT - 7.25e24T^{2} \)
31 \( 1 + 5.36e12T + 2.25e25T^{2} \)
37 \( 1 - 1.92e13iT - 4.56e26T^{2} \)
41 \( 1 - 5.42e13T + 2.61e27T^{2} \)
43 \( 1 + 3.96e13iT - 5.87e27T^{2} \)
47 \( 1 + 4.48e13T + 2.66e28T^{2} \)
53 \( 1 - 7.81e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.07e15iT - 1.27e30T^{2} \)
61 \( 1 + 1.92e15iT - 2.24e30T^{2} \)
67 \( 1 - 3.68e15iT - 1.10e31T^{2} \)
71 \( 1 - 1.02e16T + 2.96e31T^{2} \)
73 \( 1 - 1.25e16T + 4.74e31T^{2} \)
79 \( 1 + 3.24e15T + 1.81e32T^{2} \)
83 \( 1 + 4.28e15iT - 4.21e32T^{2} \)
89 \( 1 - 6.95e16T + 1.37e33T^{2} \)
97 \( 1 + 7.74e16T + 5.95e33T^{2} \)
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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

−12.84561659019573835484324316692, −12.23301237103924005197396621832, −10.69705034476509493077818598018, −9.333642987280508693475135825628, −8.000793689893638937873409606091, −6.96443933476510083009594521153, −5.10349678876547229643644225865, −4.22858921271517548443292702327, −1.93834115564346669326495606643, −1.23799980616389047684637674831, 0.67762483072906613257243421209, 2.26754923073730297631586239975, 3.74468969251542991957793813297, 5.04850074834299662404092541331, 6.59169551028921453572012027946, 7.916795137952884307434364050073, 9.246416137131090768105514958129, 10.84889156051182535530616423756, 11.26065962025572170710428474073, 13.12690563390111979758736430825

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