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Properties

Label 2-4032-48.35-c1-0-35
Degree $2$
Conductor $4032$
Sign $-0.642 + 0.766i$
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  + (−0.871 − 0.871i)5-s + 7-s + (−0.987 + 0.987i)11-s + (0.526 + 0.526i)13-s + 5.93i·17-s + (−2.78 + 2.78i)19-s − 8.59i·23-s − 3.48i·25-s + (−5.07 + 5.07i)29-s − 7.72i·31-s + (−0.871 − 0.871i)35-s + (3.36 − 3.36i)37-s − 4.55·41-s + (−1.26 − 1.26i)43-s − 5.02·47-s + ⋯
L(s)  = 1  + (−0.389 − 0.389i)5-s + 0.377·7-s + (−0.297 + 0.297i)11-s + (0.146 + 0.146i)13-s + 1.43i·17-s + (−0.638 + 0.638i)19-s − 1.79i·23-s − 0.696i·25-s + (−0.942 + 0.942i)29-s − 1.38i·31-s + (−0.147 − 0.147i)35-s + (0.553 − 0.553i)37-s − 0.711·41-s + (−0.193 − 0.193i)43-s − 0.733·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.642 + 0.766i$
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: \((2,\ 4032,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7524541708\)
\(L(\frac12)\) \(\approx\) \(0.7524541708\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.871 + 0.871i)T + 5iT^{2} \)
11 \( 1 + (0.987 - 0.987i)T - 11iT^{2} \)
13 \( 1 + (-0.526 - 0.526i)T + 13iT^{2} \)
17 \( 1 - 5.93iT - 17T^{2} \)
19 \( 1 + (2.78 - 2.78i)T - 19iT^{2} \)
23 \( 1 + 8.59iT - 23T^{2} \)
29 \( 1 + (5.07 - 5.07i)T - 29iT^{2} \)
31 \( 1 + 7.72iT - 31T^{2} \)
37 \( 1 + (-3.36 + 3.36i)T - 37iT^{2} \)
41 \( 1 + 4.55T + 41T^{2} \)
43 \( 1 + (1.26 + 1.26i)T + 43iT^{2} \)
47 \( 1 + 5.02T + 47T^{2} \)
53 \( 1 + (-2.07 - 2.07i)T + 53iT^{2} \)
59 \( 1 + (-6.52 + 6.52i)T - 59iT^{2} \)
61 \( 1 + (-8.74 - 8.74i)T + 61iT^{2} \)
67 \( 1 + (-6.20 + 6.20i)T - 67iT^{2} \)
71 \( 1 + 1.72iT - 71T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 + 5.03iT - 79T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 + 2.97T + 89T^{2} \)
97 \( 1 + 8.71T + 97T^{2} \)
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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

−8.340692358553528015542077740313, −7.61455669577548239351043679951, −6.63390262617572299286156541470, −6.02034492174261828185325341286, −5.12058515393459622934602593900, −4.24763682197340691191403383788, −3.82072803679558935396635516040, −2.45115344282310769456799383898, −1.64909654645904777766753946845, −0.22161430797063674850880373148, 1.21517814705950798740806493860, 2.45380376320737649997818315414, 3.27616164576644884014130445872, 4.05533224776761001575930189247, 5.16563171761255510352792398846, 5.49702535962546156421682529286, 6.77645152487995771441104025157, 7.16273927069975531578427941358, 7.979455562262801731815092035944, 8.578009075564705142464668095394

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