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Properties

Label 2-4032-48.35-c1-0-15
Degree $2$
Conductor $4032$
Sign $0.911 + 0.410i$
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  + (−2.28 − 2.28i)5-s + 7-s + (−3.92 + 3.92i)11-s + (1.38 + 1.38i)13-s − 4.10i·17-s + (−3.60 + 3.60i)19-s − 2.68i·23-s + 5.48i·25-s + (6.10 − 6.10i)29-s + 5.45i·31-s + (−2.28 − 2.28i)35-s + (−4.04 + 4.04i)37-s + 7.75·41-s + (−0.0577 − 0.0577i)43-s + 1.70·47-s + ⋯
L(s)  = 1  + (−1.02 − 1.02i)5-s + 0.377·7-s + (−1.18 + 1.18i)11-s + (0.385 + 0.385i)13-s − 0.995i·17-s + (−0.827 + 0.827i)19-s − 0.559i·23-s + 1.09i·25-s + (1.13 − 1.13i)29-s + 0.980i·31-s + (−0.387 − 0.387i)35-s + (−0.665 + 0.665i)37-s + 1.21·41-s + (−0.00879 − 0.00879i)43-s + 0.248·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.911 + 0.410i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.214531151\)
\(L(\frac12)\) \(\approx\) \(1.214531151\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.28 + 2.28i)T + 5iT^{2} \)
11 \( 1 + (3.92 - 3.92i)T - 11iT^{2} \)
13 \( 1 + (-1.38 - 1.38i)T + 13iT^{2} \)
17 \( 1 + 4.10iT - 17T^{2} \)
19 \( 1 + (3.60 - 3.60i)T - 19iT^{2} \)
23 \( 1 + 2.68iT - 23T^{2} \)
29 \( 1 + (-6.10 + 6.10i)T - 29iT^{2} \)
31 \( 1 - 5.45iT - 31T^{2} \)
37 \( 1 + (4.04 - 4.04i)T - 37iT^{2} \)
41 \( 1 - 7.75T + 41T^{2} \)
43 \( 1 + (0.0577 + 0.0577i)T + 43iT^{2} \)
47 \( 1 - 1.70T + 47T^{2} \)
53 \( 1 + (-0.517 - 0.517i)T + 53iT^{2} \)
59 \( 1 + (7.10 - 7.10i)T - 59iT^{2} \)
61 \( 1 + (-2.19 - 2.19i)T + 61iT^{2} \)
67 \( 1 + (-6.98 + 6.98i)T - 67iT^{2} \)
71 \( 1 + 0.356iT - 71T^{2} \)
73 \( 1 + 7.05iT - 73T^{2} \)
79 \( 1 - 0.163iT - 79T^{2} \)
83 \( 1 + (-7.33 - 7.33i)T + 83iT^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 - 8.56T + 97T^{2} \)
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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

−8.329004429690214641920880261281, −7.78113869110220718905027174584, −7.15085377296940620587168893217, −6.17976456285287951134758293229, −5.09645029390160011088380116891, −4.61682433171458382294449803969, −4.09981287402711603876972401016, −2.84874538133909583217222170240, −1.85038420345834776253933992262, −0.59856317674859359944620285544, 0.63685976409194705776759163333, 2.26387144126102984003746117364, 3.13242430970998076224675592792, 3.71153351761613885599440855286, 4.65573396870081464759664906723, 5.62497976868545548778768671253, 6.28812625872244645822348378251, 7.15708572567075064701535067992, 7.81718047715261333017259012325, 8.345975207312740305355569617583

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