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Properties

Label 2-4032-48.35-c1-0-8
Degree $2$
Conductor $4032$
Sign $0.456 - 0.889i$
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  + (−0.925 − 0.925i)5-s − 7-s + (−1.72 + 1.72i)11-s + (0.328 + 0.328i)13-s − 2.34i·17-s + (1.77 − 1.77i)19-s + 6.17i·23-s − 3.28i·25-s + (0.122 − 0.122i)29-s − 1.74i·31-s + (0.925 + 0.925i)35-s + (−1.68 + 1.68i)37-s − 2.88·41-s + (−2.77 − 2.77i)43-s + 5.92·47-s + ⋯
L(s)  = 1  + (−0.413 − 0.413i)5-s − 0.377·7-s + (−0.519 + 0.519i)11-s + (0.0910 + 0.0910i)13-s − 0.567i·17-s + (0.408 − 0.408i)19-s + 1.28i·23-s − 0.657i·25-s + (0.0227 − 0.0227i)29-s − 0.313i·31-s + (0.156 + 0.156i)35-s + (−0.276 + 0.276i)37-s − 0.451·41-s + (−0.422 − 0.422i)43-s + 0.863·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.456 - 0.889i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.095343602\)
\(L(\frac12)\) \(\approx\) \(1.095343602\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.925 + 0.925i)T + 5iT^{2} \)
11 \( 1 + (1.72 - 1.72i)T - 11iT^{2} \)
13 \( 1 + (-0.328 - 0.328i)T + 13iT^{2} \)
17 \( 1 + 2.34iT - 17T^{2} \)
19 \( 1 + (-1.77 + 1.77i)T - 19iT^{2} \)
23 \( 1 - 6.17iT - 23T^{2} \)
29 \( 1 + (-0.122 + 0.122i)T - 29iT^{2} \)
31 \( 1 + 1.74iT - 31T^{2} \)
37 \( 1 + (1.68 - 1.68i)T - 37iT^{2} \)
41 \( 1 + 2.88T + 41T^{2} \)
43 \( 1 + (2.77 + 2.77i)T + 43iT^{2} \)
47 \( 1 - 5.92T + 47T^{2} \)
53 \( 1 + (-0.973 - 0.973i)T + 53iT^{2} \)
59 \( 1 + (8.33 - 8.33i)T - 59iT^{2} \)
61 \( 1 + (-4.28 - 4.28i)T + 61iT^{2} \)
67 \( 1 + (1.78 - 1.78i)T - 67iT^{2} \)
71 \( 1 - 8.57iT - 71T^{2} \)
73 \( 1 - 6.41iT - 73T^{2} \)
79 \( 1 + 5.38iT - 79T^{2} \)
83 \( 1 + (3.46 + 3.46i)T + 83iT^{2} \)
89 \( 1 - 1.51T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−8.671556511657917890099776650292, −7.64768175467788615355060336690, −7.33221008111893547621695320005, −6.39098573124297855701697317599, −5.49694716085821523324045884347, −4.84085991779562362013044504600, −4.02150285165938446986112274756, −3.13706846273051825200657031041, −2.20101526223365760319930356302, −0.911424531986132050798148823331, 0.38829300399291091391476745898, 1.82740014960597603546727538636, 3.00484682531933031535988549343, 3.50716079167165528366342249811, 4.49313415183184641640879282628, 5.39522380284727654051657525676, 6.16425183094287576634562215009, 6.83103375258587685795307736425, 7.63980139773891728036235134130, 8.258679742890440225299549991523

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