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Properties

Label 2-4032-56.27-c1-0-26
Degree $2$
Conductor $4032$
Sign $0.132 - 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.132 - 0.991i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.132 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.823549258\)
\(L(\frac12)\) \(\approx\) \(1.823549258\)
\(L(\frac{3}{2})\) 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

−8.656170103031154407126781912175, −8.000040421771221475339482890762, −6.81191024539296255481758566271, −6.42837466368096252564109590834, −5.65044564104554393952791140034, −5.17075927518912529681949697180, −3.99307716629073561587293486102, −2.90455206861621810106627894794, −2.29981738220395457107883664282, −1.26224546481457221163748071518, 0.49997052174288443335197595928, 1.91279954344142854854824916924, 2.65894391610923854749288908417, 3.36958761822910918055150164848, 4.88256290314134665415895712862, 5.16723006328739893471090107003, 6.27325479238664367007332130307, 6.57111337242799907876585569762, 7.34829073688799959545371871863, 8.465561391061210023980066694126

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