Properties

Label 2-4032-21.20-c1-0-0
Degree $2$
Conductor $4032$
Sign $-0.992 - 0.119i$
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  − 2.61·5-s + (1.25 + 2.32i)7-s − 4.29i·11-s − 1.53i·13-s − 0.448·17-s − 1.92i·19-s + 0.737i·23-s + 1.82·25-s + 1.41i·29-s + 6.58i·31-s + (−3.29 − 6.08i)35-s + 2.82·37-s − 6.94·41-s + 9.64·43-s + 2.72·47-s + ⋯
L(s)  = 1  − 1.16·5-s + (0.475 + 0.879i)7-s − 1.29i·11-s − 0.424i·13-s − 0.108·17-s − 0.442i·19-s + 0.153i·23-s + 0.365·25-s + 0.262i·29-s + 1.18i·31-s + (−0.556 − 1.02i)35-s + 0.464·37-s − 1.08·41-s + 1.47·43-s + 0.397·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.992 - 0.119i$
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: \((2,\ 4032,\ (\ :1/2),\ -0.992 - 0.119i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07957796095\)
\(L(\frac12)\) \(\approx\) \(0.07957796095\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 2.32i)T \)
good5 \( 1 + 2.61T + 5T^{2} \)
11 \( 1 + 4.29iT - 11T^{2} \)
13 \( 1 + 1.53iT - 13T^{2} \)
17 \( 1 + 0.448T + 17T^{2} \)
19 \( 1 + 1.92iT - 19T^{2} \)
23 \( 1 - 0.737iT - 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 - 6.58iT - 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + 6.94T + 41T^{2} \)
43 \( 1 - 9.64T + 43T^{2} \)
47 \( 1 - 2.72T + 47T^{2} \)
53 \( 1 - 2.58iT - 53T^{2} \)
59 \( 1 + 9.30T + 59T^{2} \)
61 \( 1 - 8.28iT - 61T^{2} \)
67 \( 1 + 1.47T + 67T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + 11.0iT - 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 - 6.75iT - 97T^{2} \)
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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

−8.916086001272998545026178495856, −7.972518012590313821938974927866, −7.62383861221604952109858917719, −6.56457884852220015714429197085, −5.78183128552835864363709507361, −5.08292001558518245979913847785, −4.22673812317621930165875306111, −3.31610970166252265179517992935, −2.69040841789614316498262179578, −1.26220309079315162638512492426, 0.02483536596207907528931513678, 1.38037805110527646075416450326, 2.45129469781181605214563694219, 3.75137938937165008492785657815, 4.24140031050151537900434209380, 4.76196142802270088100657670841, 5.89928956847141290167404814562, 6.94060049104742882701275839938, 7.39471592941580932677720023399, 7.934140985855386614254342459398

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