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Properties

Label 2-4032-56.27-c1-0-67
Degree $2$
Conductor $4032$
Sign $0.707 + 0.707i$
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  + 3.91·5-s − 2.64i·7-s + 5.59·13-s − 8.66i·19-s + 6i·23-s + 10.2·25-s − 10.3i·35-s − 7.00·49-s + 5.29i·59-s − 0.543·61-s + 21.8·65-s − 15.8i·71-s − 5.29i·79-s + 18.1i·83-s − 14.8i·91-s + ⋯
L(s)  = 1  + 1.74·5-s − 0.999i·7-s + 1.55·13-s − 1.98i·19-s + 1.25i·23-s + 2.05·25-s − 1.74i·35-s − 49-s + 0.689i·59-s − 0.0695·61-s + 2.71·65-s − 1.88i·71-s − 0.595i·79-s + 1.99i·83-s − 1.55i·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.707 + 0.707i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.021934852\)
\(L(\frac12)\) \(\approx\) \(3.021934852\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64iT \)
good5 \( 1 - 3.91T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5.59T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8.66iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 + 0.543T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 5.29iT - 79T^{2} \)
83 \( 1 - 18.1iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.525750480570031689978972290292, −7.48260431948526184019157318852, −6.73342766047465101904995669991, −6.20868889592101669890749175198, −5.43769647582477033768787097758, −4.70280738173816772429367454724, −3.67456761778818602736027602953, −2.78217788162723537679936471164, −1.70597479235436982718858655651, −0.934455447373734562229811229822, 1.33105211567630243131033988273, 2.00166641622371586198774981757, 2.88279871742941344479839719741, 3.88506136453088209936097966287, 5.03818147148431219165409555353, 5.88544199367342488282577277447, 6.01673059253701563223213855309, 6.75039509705992944496955748853, 8.090742922103545718635627875703, 8.611951713162222336324842909723

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