Properties

Label 2-4032-56.27-c1-0-0
Degree $2$
Conductor $4032$
Sign $-0.924 + 0.380i$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.88·5-s + (−2.62 + 0.339i)7-s + 1.48·11-s + 2·13-s + 3.76i·17-s − 0.679i·19-s + 5.20i·23-s + 10.0·25-s + 7.03i·29-s + 6.42·31-s + (10.1 − 1.32i)35-s − 8.71i·37-s − 3.16i·41-s + 8.20·43-s − 9.88·47-s + ⋯
L(s)  = 1  − 1.73·5-s + (−0.991 + 0.128i)7-s + 0.446·11-s + 0.554·13-s + 0.913i·17-s − 0.155i·19-s + 1.08i·23-s + 2.01·25-s + 1.30i·29-s + 1.15·31-s + (1.72 − 0.223i)35-s − 1.43i·37-s − 0.493i·41-s + 1.25·43-s − 1.44·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.924 + 0.380i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.924 + 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02178647919\)
\(L(\frac12)\) \(\approx\) \(0.02178647919\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (2.62 - 0.339i)T \)
good5 \( 1 + 3.88T + 5T^{2} \)
11 \( 1 - 1.48T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 3.76iT - 17T^{2} \)
19 \( 1 + 0.679iT - 19T^{2} \)
23 \( 1 - 5.20iT - 23T^{2} \)
29 \( 1 - 7.03iT - 29T^{2} \)
31 \( 1 - 6.42T + 31T^{2} \)
37 \( 1 + 8.71iT - 37T^{2} \)
41 \( 1 + 3.16iT - 41T^{2} \)
43 \( 1 - 8.20T + 43T^{2} \)
47 \( 1 + 9.88T + 47T^{2} \)
53 \( 1 + 6.42iT - 53T^{2} \)
59 \( 1 + 7.76iT - 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 8.81T + 67T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 + 4.44iT - 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 - 9.88iT - 97T^{2} \)
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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

−8.796498695137236545107578536386, −8.100582570307051708259327459117, −7.41063587141254230967440910765, −6.75241104461477051687020954227, −6.04283344151542817904053195886, −5.04156381725785124434605020943, −3.91441093015374427705387410004, −3.72557415795863154847411659594, −2.83340995355514430512361160642, −1.27083299168847145997356186623, 0.008426563678264835924467125543, 0.982930203167179147124353260843, 2.81170078779686062408555210434, 3.30206239418491145113301103494, 4.33101816564393559303695480387, 4.58499503913830490280400565479, 6.10491928057944240126850024439, 6.54354846834221362658807253593, 7.40747497620712218202576729455, 7.963829428947653981060400361417

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