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Properties

Label 2-4032-48.11-c1-0-28
Degree $2$
Conductor $4032$
Sign $0.962 - 0.271i$
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  + (−2.98 + 2.98i)5-s + 7-s + (2.32 + 2.32i)11-s + (1.27 − 1.27i)13-s − 3.56i·17-s + (0.796 + 0.796i)19-s − 1.75i·23-s − 12.8i·25-s + (−1.87 − 1.87i)29-s − 7.15i·31-s + (−2.98 + 2.98i)35-s + (4.64 + 4.64i)37-s + 8.98·41-s + (6.04 − 6.04i)43-s − 6.99·47-s + ⋯
L(s)  = 1  + (−1.33 + 1.33i)5-s + 0.377·7-s + (0.701 + 0.701i)11-s + (0.354 − 0.354i)13-s − 0.863i·17-s + (0.182 + 0.182i)19-s − 0.366i·23-s − 2.57i·25-s + (−0.348 − 0.348i)29-s − 1.28i·31-s + (−0.505 + 0.505i)35-s + (0.762 + 0.762i)37-s + 1.40·41-s + (0.921 − 0.921i)43-s − 1.01·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.509008059\)
\(L(\frac12)\) \(\approx\) \(1.509008059\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.98 - 2.98i)T - 5iT^{2} \)
11 \( 1 + (-2.32 - 2.32i)T + 11iT^{2} \)
13 \( 1 + (-1.27 + 1.27i)T - 13iT^{2} \)
17 \( 1 + 3.56iT - 17T^{2} \)
19 \( 1 + (-0.796 - 0.796i)T + 19iT^{2} \)
23 \( 1 + 1.75iT - 23T^{2} \)
29 \( 1 + (1.87 + 1.87i)T + 29iT^{2} \)
31 \( 1 + 7.15iT - 31T^{2} \)
37 \( 1 + (-4.64 - 4.64i)T + 37iT^{2} \)
41 \( 1 - 8.98T + 41T^{2} \)
43 \( 1 + (-6.04 + 6.04i)T - 43iT^{2} \)
47 \( 1 + 6.99T + 47T^{2} \)
53 \( 1 + (-0.536 + 0.536i)T - 53iT^{2} \)
59 \( 1 + (-0.119 - 0.119i)T + 59iT^{2} \)
61 \( 1 + (-10.9 + 10.9i)T - 61iT^{2} \)
67 \( 1 + (3.83 + 3.83i)T + 67iT^{2} \)
71 \( 1 - 9.96iT - 71T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 - 15.4iT - 79T^{2} \)
83 \( 1 + (4.23 - 4.23i)T - 83iT^{2} \)
89 \( 1 - 2.38T + 89T^{2} \)
97 \( 1 - 6.00T + 97T^{2} \)
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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

−8.040259497475198852042169374443, −7.87683880334498322511349106457, −6.99538710400992719489323832868, −6.55802096069283318954039204715, −5.55461051205441262602978930838, −4.36671815886341996649724694925, −3.97584741533253079998670999580, −3.03073969715574374368456870459, −2.23695030278452273450329260389, −0.64486056911219672974833997985, 0.828967111041408128760043868561, 1.55590834485573809159735160940, 3.17888342401680689187734927294, 4.02661260610658666215854202012, 4.41374354598278596416644564023, 5.37069394880093667732094006574, 6.08174792195395110782237948792, 7.16627530931842013519700199070, 7.79393500779972791899273228080, 8.520646291234570454265326278025

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