Properties

Label 2-4032-56.27-c1-0-43
Degree $2$
Conductor $4032$
Sign $0.899 - 0.436i$
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  − 2.82·5-s + (1.73 + 2i)7-s + 4.89·11-s + 6·13-s + 4.89i·17-s − 8.48i·23-s + 3.00·25-s + 3.46·31-s + (−4.89 − 5.65i)35-s − 6.92i·37-s + 4.89i·41-s + 3.46·43-s − 9.79·47-s + (−1.00 + 6.92i)49-s − 9.79i·53-s + ⋯
L(s)  = 1  − 1.26·5-s + (0.654 + 0.755i)7-s + 1.47·11-s + 1.66·13-s + 1.18i·17-s − 1.76i·23-s + 0.600·25-s + 0.622·31-s + (−0.828 − 0.956i)35-s − 1.13i·37-s + 0.765i·41-s + 0.528·43-s − 1.42·47-s + (−0.142 + 0.989i)49-s − 1.34i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.899 - 0.436i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.899 - 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.960601503\)
\(L(\frac12)\) \(\approx\) \(1.960601503\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 2i)T \)
good5 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 8.48iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 - 4.89iT - 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + 9.79iT - 53T^{2} \)
59 \( 1 - 5.65iT - 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 5.65iT - 83T^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−8.409664255378804194848969647556, −8.148267520113372026129049541823, −6.92151456466506536253213646152, −6.35122768172717986066038954866, −5.64516669613222469702800910403, −4.35101310397607466443180067991, −4.08329015168181078216153306684, −3.22852104590646133007663059317, −1.91571470477267452135372930172, −0.921576956845613378767616426951, 0.827658362095883968433396681585, 1.55713160420239701321147716183, 3.30922860052487716512540987952, 3.76694195084383060561113508660, 4.39814554843849250266046777002, 5.28594014307674244328177339453, 6.40793867620285264904382613332, 6.95382367003710302094031943047, 7.79617666609163259047837943755, 8.182513156264960756383676588198

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