Properties

Label 2-4032-1.1-c1-0-22
Degree $2$
Conductor $4032$
Sign $1$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 4·11-s + 4·13-s + 2·17-s − 6·19-s + 8·23-s − 5·25-s + 2·29-s + 4·31-s − 10·37-s + 10·41-s + 4·43-s + 4·47-s + 49-s − 2·53-s − 10·59-s + 8·61-s − 8·67-s − 6·73-s + 4·77-s + 16·79-s − 2·83-s − 18·89-s + 4·91-s − 2·97-s + 4·103-s + 16·107-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.20·11-s + 1.10·13-s + 0.485·17-s − 1.37·19-s + 1.66·23-s − 25-s + 0.371·29-s + 0.718·31-s − 1.64·37-s + 1.56·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.274·53-s − 1.30·59-s + 1.02·61-s − 0.977·67-s − 0.702·73-s + 0.455·77-s + 1.80·79-s − 0.219·83-s − 1.90·89-s + 0.419·91-s − 0.203·97-s + 0.394·103-s + 1.54·107-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4032} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.372477485\)
\(L(\frac12)\) \(\approx\) \(2.372477485\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.623707715056393928855156330414, −7.74725258589973907924089231007, −6.89113023940731430405555362979, −6.26792363571132101849104139509, −5.58346395686892548609612901812, −4.50652786046145735328701056321, −3.94318918304103393410797545674, −3.02875714210021469734915733866, −1.82093855863452524537562095799, −0.948205382819290662741350247887, 0.948205382819290662741350247887, 1.82093855863452524537562095799, 3.02875714210021469734915733866, 3.94318918304103393410797545674, 4.50652786046145735328701056321, 5.58346395686892548609612901812, 6.26792363571132101849104139509, 6.89113023940731430405555362979, 7.74725258589973907924089231007, 8.623707715056393928855156330414

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