Properties

Label 2-4032-28.27-c1-0-17
Degree $2$
Conductor $4032$
Sign $-i$
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  + 2.64i·7-s − 4i·11-s + 8i·23-s + 5·25-s − 10.5·29-s + 6·37-s + 5.29i·43-s − 7.00·49-s + 10.5·53-s + 15.8i·67-s − 16i·71-s + 10.5·77-s + 15.8i·79-s + 20i·107-s − 18·109-s + ⋯
L(s)  = 1  + 0.999i·7-s − 1.20i·11-s + 1.66i·23-s + 25-s − 1.96·29-s + 0.986·37-s + 0.806i·43-s − 49-s + 1.45·53-s + 1.93i·67-s − 1.89i·71-s + 1.20·77-s + 1.78i·79-s + 1.93i·107-s − 1.72·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428927337\)
\(L(\frac12)\) \(\approx\) \(1.428927337\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64iT \)
good5 \( 1 - 5T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 10.5T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 5.29iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 + 16iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.718782811869008928280753253005, −7.935806734056698757995056245383, −7.24983385463126012597621684221, −6.23175803978641084918528897380, −5.66111272808665735256117518001, −5.12423103226544268649342374279, −3.89767791402347863241630828830, −3.18181539933765717095312861824, −2.30020590929615242346956981262, −1.13469035014923703245647784304, 0.44227023936944190438865429585, 1.70463352289746863237918581525, 2.65553400733955010443916488962, 3.80985509520358652365013434177, 4.41307807814690098082746382415, 5.12305601928994183347257562702, 6.15700409157034182000590658016, 7.00654267519237419057726562349, 7.34233595291588432134201943112, 8.194943617996727042326131941446

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