Properties

Label 2-4032-56.27-c1-0-57
Degree $2$
Conductor $4032$
Sign $0.902 + 0.430i$
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  + 2.76·5-s + (0.480 − 2.60i)7-s + 5.75·11-s + 2·13-s − 6.71i·17-s + 5.20i·19-s + 4.43i·23-s + 2.66·25-s + 1.54i·29-s + 8.05·31-s + (1.33 − 7.20i)35-s + 4.42i·37-s + 0.209i·41-s + 10.5·43-s − 4.58·47-s + ⋯
L(s)  = 1  + 1.23·5-s + (0.181 − 0.983i)7-s + 1.73·11-s + 0.554·13-s − 1.62i·17-s + 1.19i·19-s + 0.924i·23-s + 0.533·25-s + 0.286i·29-s + 1.44·31-s + (0.225 − 1.21i)35-s + 0.727i·37-s + 0.0326i·41-s + 1.60·43-s − 0.669·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.055323555\)
\(L(\frac12)\) \(\approx\) \(3.055323555\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.480 + 2.60i)T \)
good5 \( 1 - 2.76T + 5T^{2} \)
11 \( 1 - 5.75T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 6.71iT - 17T^{2} \)
19 \( 1 - 5.20iT - 19T^{2} \)
23 \( 1 - 4.43iT - 23T^{2} \)
29 \( 1 - 1.54iT - 29T^{2} \)
31 \( 1 - 8.05T + 31T^{2} \)
37 \( 1 - 4.42iT - 37T^{2} \)
41 \( 1 - 0.209iT - 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 4.58T + 47T^{2} \)
53 \( 1 + 8.05iT - 53T^{2} \)
59 \( 1 - 5.53iT - 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 4.04T + 67T^{2} \)
71 \( 1 + 1.10iT - 71T^{2} \)
73 \( 1 - 9.59iT - 73T^{2} \)
79 \( 1 - 14.7iT - 79T^{2} \)
83 \( 1 + 8.86iT - 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 - 4.58iT - 97T^{2} \)
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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

−8.463034884463876510790450798828, −7.55793862395211532620706616148, −6.80602281594299158873660700170, −6.25294193943978628805137485050, −5.52992437025702896528866324126, −4.55979139775842548227559752616, −3.83440025253934210447415831696, −2.91232478097438326169409927914, −1.59936073333356926646443002196, −1.09008592049863520992888098206, 1.19369671947814296886386512867, 1.98667923812429529081835840817, 2.82147308109784252436732539847, 3.99824513835309935930150172707, 4.71974946907942806512336193211, 5.84040039649677392092356564294, 6.21931281894246841888916432648, 6.63954204640471995706439672570, 7.927387467367870148362239123379, 8.768920318675575934150550537148

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