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Properties

Label 2-4032-24.11-c1-0-19
Degree $2$
Conductor $4032$
Sign $0.346 - 0.938i$
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  + 1.51·5-s + i·7-s + 0.935i·11-s + 2.14i·13-s + 2.62i·17-s + 3.46·19-s + 6.86·23-s − 2.70·25-s − 2.44·29-s − 2i·31-s + 1.51i·35-s − 2.14i·37-s + 2.62i·41-s + 7.74·43-s − 49-s + ⋯
L(s)  = 1  + 0.677·5-s + 0.377i·7-s + 0.282i·11-s + 0.593i·13-s + 0.635i·17-s + 0.794·19-s + 1.43·23-s − 0.541·25-s − 0.454·29-s − 0.359i·31-s + 0.255i·35-s − 0.351i·37-s + 0.409i·41-s + 1.18·43-s − 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.346 - 0.938i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.346 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.062768433\)
\(L(\frac12)\) \(\approx\) \(2.062768433\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 - 1.51T + 5T^{2} \)
11 \( 1 - 0.935iT - 11T^{2} \)
13 \( 1 - 2.14iT - 13T^{2} \)
17 \( 1 - 2.62iT - 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 6.86T + 23T^{2} \)
29 \( 1 + 2.44T + 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + 2.14iT - 37T^{2} \)
41 \( 1 - 2.62iT - 41T^{2} \)
43 \( 1 - 7.74T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 9.79iT - 59T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
67 \( 1 + 2.14T + 67T^{2} \)
71 \( 1 - 1.62T + 71T^{2} \)
73 \( 1 + 5.70T + 73T^{2} \)
79 \( 1 - 1.70iT - 79T^{2} \)
83 \( 1 - 1.87iT - 83T^{2} \)
89 \( 1 + 2.62iT - 89T^{2} \)
97 \( 1 - 17.7T + 97T^{2} \)
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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

−8.821931818757198563250061257550, −7.73521316418462595223663978106, −7.18412480054098767402179003474, −6.23712558939223708513283288964, −5.71704436154987685592092344638, −4.90004855467176682313943530714, −4.05174018364879659245222243735, −3.02089017638425213377967764903, −2.14110615463576545867379936949, −1.21864376071694399695074921029, 0.62623330916934056076784980731, 1.71075469924936742855866914746, 2.86936861891064807860939362420, 3.49506479229246709297147901618, 4.69160200548562746552928492806, 5.32370921048655158686176127527, 6.03013242683335797758783559127, 6.87781854366743453621944717578, 7.54181069021091760799367823367, 8.238275857804298779734780522710

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