Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.639 + 0.769i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.646·5-s + i·7-s − 3.09i·11-s + 0.913i·13-s + 6.77i·17-s − 5.29·19-s + 2.53·23-s − 4.58·25-s − 9.93·29-s − 9.16i·31-s + 0.646i·35-s − 7.84i·37-s − 9.01i·41-s + 12.2·43-s − 5.65·47-s + ⋯
L(s)  = 1  + 0.288·5-s + 0.377i·7-s − 0.933i·11-s + 0.253i·13-s + 1.64i·17-s − 1.21·19-s + 0.528·23-s − 0.916·25-s − 1.84·29-s − 1.64i·31-s + 0.109i·35-s − 1.28i·37-s − 1.40i·41-s + 1.86·43-s − 0.825·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.639 + 0.769i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.639 + 0.769i)$
$L(1)$  $\approx$  $0.6966156387$
$L(\frac12)$  $\approx$  $0.6966156387$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 0.646T + 5T^{2} \)
11 \( 1 + 3.09iT - 11T^{2} \)
13 \( 1 - 0.913iT - 13T^{2} \)
17 \( 1 - 6.77iT - 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 - 2.53T + 23T^{2} \)
29 \( 1 + 9.93T + 29T^{2} \)
31 \( 1 + 9.16iT - 31T^{2} \)
37 \( 1 + 7.84iT - 37T^{2} \)
41 \( 1 + 9.01iT - 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 1.15T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 + 5.36T + 71T^{2} \)
73 \( 1 + 0.417T + 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 + 15.9iT - 83T^{2} \)
89 \( 1 - 9.01iT - 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.175336157242274073580008369034, −7.59593465144466780710787490773, −6.53435893307611474833445545870, −5.83941185018851408859864939784, −5.53961408768790835428854119231, −4.06181116936448422049998995701, −3.80360536159668173692672699252, −2.40856050812174339877273902988, −1.77561053162430344545300296820, −0.19018907050634605734134476642, 1.30953129224655850211064709490, 2.33079324876759495785543143678, 3.21446846022159707667357579574, 4.29088887866180667353244614219, 4.89493219865632289141054834959, 5.70170264711434742441284295696, 6.66003602606713845253503497965, 7.20215241438318370304661948790, 7.86570891407413334116174257443, 8.761454829440315427312026514266

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