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Properties

Label 2-4032-168.125-c1-0-36
Degree $2$
Conductor $4032$
Sign $0.530 + 0.847i$
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  + 1.21i·5-s + (0.355 − 2.62i)7-s − 4.06·11-s − 1.42·13-s + 4.74·17-s − 4.84·19-s + 6.03i·23-s + 3.51·25-s + 3.45·29-s + 3.15i·31-s + (3.19 + 0.432i)35-s − 9.24i·37-s + 1.24·41-s + 3.23i·43-s + 9.43·47-s + ⋯
L(s)  = 1  + 0.544i·5-s + (0.134 − 0.990i)7-s − 1.22·11-s − 0.396·13-s + 1.15·17-s − 1.11·19-s + 1.25i·23-s + 0.703·25-s + 0.641·29-s + 0.566i·31-s + (0.539 + 0.0731i)35-s − 1.51i·37-s + 0.194·41-s + 0.492i·43-s + 1.37·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.434268288\)
\(L(\frac12)\) \(\approx\) \(1.434268288\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.355 + 2.62i)T \)
good5 \( 1 - 1.21iT - 5T^{2} \)
11 \( 1 + 4.06T + 11T^{2} \)
13 \( 1 + 1.42T + 13T^{2} \)
17 \( 1 - 4.74T + 17T^{2} \)
19 \( 1 + 4.84T + 19T^{2} \)
23 \( 1 - 6.03iT - 23T^{2} \)
29 \( 1 - 3.45T + 29T^{2} \)
31 \( 1 - 3.15iT - 31T^{2} \)
37 \( 1 + 9.24iT - 37T^{2} \)
41 \( 1 - 1.24T + 41T^{2} \)
43 \( 1 - 3.23iT - 43T^{2} \)
47 \( 1 - 9.43T + 47T^{2} \)
53 \( 1 + 2.44T + 53T^{2} \)
59 \( 1 + 10.2iT - 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 6.51iT - 67T^{2} \)
71 \( 1 + 2.14iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 16.3iT - 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−8.085598237381979544238442776858, −7.56060788020354588150146265283, −7.05787132040339096762965221625, −6.13354560442116766508118975501, −5.30267090354864256784411833165, −4.58015689789721101106393171726, −3.60030621801849887077573732413, −2.90233398574638139281064061473, −1.83882398694555259882142387662, −0.49010178071526457134561434641, 0.950879374968441261971407955209, 2.38786959773250035146782887473, 2.77402292257228777693892385683, 4.13502749076548621048911394130, 4.95948024541545441286618104709, 5.45936408255970408284710333930, 6.24845504453552974111914531063, 7.12776386070512442807257280780, 8.143068974589510435096603554811, 8.398728625771216576295820296725

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