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Properties

Label 2-4032-168.125-c1-0-44
Degree $2$
Conductor $4032$
Sign $0.883 - 0.469i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.883 - 0.469i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.883 - 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.435285647\)
\(L(\frac12)\) \(\approx\) \(2.435285647\)
\(L(\frac{3}{2})\) 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

−8.613494759743754908413231335005, −7.70564182224949850267971777308, −7.06555020651913434433264044904, −6.14650333208251944664393883032, −5.74168009474076050566101851294, −4.77135148325057039801585708333, −3.76813821912976321113167437738, −3.05481793755788756082774378871, −2.11158893846503755917134064355, −0.966473481379860458475961357231, 1.06179357415606587798383882345, 1.40641519143343020593230177671, 3.16875210631507286937991247586, 3.73338529049342558949952821059, 4.57108736319375889514479656687, 5.39500962127271932603658858794, 6.12627979833975951911087238808, 7.16291057208991552182815195554, 7.47794693314468580643479740531, 8.419354614953028762532705072526

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