Properties

Label 2-4032-1.1-c1-0-19
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  − 2·5-s + 7-s + 4·11-s + 6·13-s + 2·17-s + 4·19-s − 4·23-s − 25-s − 2·29-s − 8·31-s − 2·35-s + 10·37-s + 2·41-s + 8·43-s + 49-s − 10·53-s − 8·55-s + 12·59-s − 10·61-s − 12·65-s − 8·67-s + 12·71-s + 2·73-s + 4·77-s − 12·83-s − 4·85-s − 6·89-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.338·35-s + 1.64·37-s + 0.312·41-s + 1.21·43-s + 1/7·49-s − 1.37·53-s − 1.07·55-s + 1.56·59-s − 1.28·61-s − 1.48·65-s − 0.977·67-s + 1.42·71-s + 0.234·73-s + 0.455·77-s − 1.31·83-s − 0.433·85-s − 0.635·89-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

\(L(1)\) \(\approx\) \(2.014469483\)
\(L(\frac12)\) \(\approx\) \(2.014469483\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−8.320383312015219226294100843388, −7.77853019762625190716075044614, −7.12476921432385806506284419378, −6.09664756569615751308461657816, −5.68030407373648152373224072684, −4.38156718899132832416834458068, −3.87697779819423025511035869420, −3.24828556873724268658050054220, −1.76295140196623459214733320395, −0.871945477605480385408643863818, 0.871945477605480385408643863818, 1.76295140196623459214733320395, 3.24828556873724268658050054220, 3.87697779819423025511035869420, 4.38156718899132832416834458068, 5.68030407373648152373224072684, 6.09664756569615751308461657816, 7.12476921432385806506284419378, 7.77853019762625190716075044614, 8.320383312015219226294100843388

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