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Properties

Label 2-4032-48.35-c1-0-14
Degree $2$
Conductor $4032$
Sign $0.938 - 0.344i$
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  + (−0.495 − 0.495i)5-s + 7-s + (−0.675 + 0.675i)11-s + (−2.39 − 2.39i)13-s + 1.03i·17-s + (−0.913 + 0.913i)19-s + 2.92i·23-s − 4.50i·25-s + (−3.39 + 3.39i)29-s + 9.43i·31-s + (−0.495 − 0.495i)35-s + (5.47 − 5.47i)37-s + 6.91·41-s + (2.30 + 2.30i)43-s + 9.68·47-s + ⋯
L(s)  = 1  + (−0.221 − 0.221i)5-s + 0.377·7-s + (−0.203 + 0.203i)11-s + (−0.664 − 0.664i)13-s + 0.251i·17-s + (−0.209 + 0.209i)19-s + 0.610i·23-s − 0.901i·25-s + (−0.629 + 0.629i)29-s + 1.69i·31-s + (−0.0836 − 0.0836i)35-s + (0.900 − 0.900i)37-s + 1.07·41-s + (0.351 + 0.351i)43-s + 1.41·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.636523803\)
\(L(\frac12)\) \(\approx\) \(1.636523803\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.495 + 0.495i)T + 5iT^{2} \)
11 \( 1 + (0.675 - 0.675i)T - 11iT^{2} \)
13 \( 1 + (2.39 + 2.39i)T + 13iT^{2} \)
17 \( 1 - 1.03iT - 17T^{2} \)
19 \( 1 + (0.913 - 0.913i)T - 19iT^{2} \)
23 \( 1 - 2.92iT - 23T^{2} \)
29 \( 1 + (3.39 - 3.39i)T - 29iT^{2} \)
31 \( 1 - 9.43iT - 31T^{2} \)
37 \( 1 + (-5.47 + 5.47i)T - 37iT^{2} \)
41 \( 1 - 6.91T + 41T^{2} \)
43 \( 1 + (-2.30 - 2.30i)T + 43iT^{2} \)
47 \( 1 - 9.68T + 47T^{2} \)
53 \( 1 + (-8.02 - 8.02i)T + 53iT^{2} \)
59 \( 1 + (1.14 - 1.14i)T - 59iT^{2} \)
61 \( 1 + (7.05 + 7.05i)T + 61iT^{2} \)
67 \( 1 + (-5.66 + 5.66i)T - 67iT^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + 6.66iT - 73T^{2} \)
79 \( 1 - 2.26iT - 79T^{2} \)
83 \( 1 + (0.619 + 0.619i)T + 83iT^{2} \)
89 \( 1 + 3.99T + 89T^{2} \)
97 \( 1 - 6.82T + 97T^{2} \)
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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

−8.485539089523702218329749453629, −7.60894120785439881215331110037, −7.32132119430453822047998669433, −6.16836484582680430602912984421, −5.48167070514849964040417933196, −4.72381256272244217620510359151, −3.98214995302217921483532952188, −2.96425933865292209004343500129, −2.05391375888591176135669100153, −0.839806468221687923702117806088, 0.62829704959818988518742135241, 2.07517202785194139096629205676, 2.74788685439949049058372048565, 3.97402469697329497940038330971, 4.48835417983084483861357645741, 5.47549492421929920777785195196, 6.14477950245263568071457401370, 7.12834615282725405394488433277, 7.57284488405768913397565161344, 8.338226545471453070550452632345

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