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Properties

Label 2-4032-21.20-c1-0-34
Degree $2$
Conductor $4032$
Sign $0.746 - 0.665i$
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  + 3.36·5-s + (2.57 + 0.595i)7-s + 1.64i·11-s − 3.06i·13-s − 3.36·17-s + 5.53i·19-s + 1.64i·23-s + 6.29·25-s + 6.06i·29-s + 4.33i·31-s + (8.66 + 2i)35-s − 7.29·37-s + 0.979·41-s + 5.15·43-s + 11.1·47-s + ⋯
L(s)  = 1  + 1.50·5-s + (0.974 + 0.224i)7-s + 0.496i·11-s − 0.851i·13-s − 0.814·17-s + 1.26i·19-s + 0.343i·23-s + 1.25·25-s + 1.12i·29-s + 0.779i·31-s + (1.46 + 0.338i)35-s − 1.19·37-s + 0.152·41-s + 0.786·43-s + 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3905, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.746 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.860777638\)
\(L(\frac12)\) \(\approx\) \(2.860777638\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.57 - 0.595i)T \)
good5 \( 1 - 3.36T + 5T^{2} \)
11 \( 1 - 1.64iT - 11T^{2} \)
13 \( 1 + 3.06iT - 13T^{2} \)
17 \( 1 + 3.36T + 17T^{2} \)
19 \( 1 - 5.53iT - 19T^{2} \)
23 \( 1 - 1.64iT - 23T^{2} \)
29 \( 1 - 6.06iT - 29T^{2} \)
31 \( 1 - 4.33iT - 31T^{2} \)
37 \( 1 + 7.29T + 37T^{2} \)
41 \( 1 - 0.979T + 41T^{2} \)
43 \( 1 - 5.15T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 - 4.24iT - 53T^{2} \)
59 \( 1 + 6.13T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 - 5.64iT - 71T^{2} \)
73 \( 1 + 8.11iT - 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 7.70T + 89T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−8.707910515258646763005219511463, −7.78370379681450040166851557925, −7.07934205451120277795837565327, −6.15367819505244552629519386927, −5.49588715449341546233033813105, −5.05113215695530877195011034547, −4.01881480595331892308758874044, −2.82777778769389098250140551673, −1.96095533646777656311702257717, −1.32542696294758125526754983067, 0.829666570875955344147672960821, 2.10151797894402207731550135418, 2.37283305107070001615275820935, 3.87572340626510169225952729962, 4.73075133441118285456167745056, 5.33134400330906792873582825682, 6.22378195501994252118951631223, 6.71051350139249148437255309164, 7.61652038690282340989676302239, 8.507158955480572556497490626162

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