Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.430 - 0.902i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.76·5-s + (−0.480 − 2.60i)7-s − 5.75·11-s + 2·13-s + 6.71i·17-s + 5.20i·19-s + 4.43i·23-s + 2.66·25-s − 1.54i·29-s − 8.05·31-s + (−1.33 − 7.20i)35-s − 4.42i·37-s − 0.209i·41-s − 10.5·43-s + 4.58·47-s + ⋯
L(s)  = 1  + 1.23·5-s + (−0.181 − 0.983i)7-s − 1.73·11-s + 0.554·13-s + 1.62i·17-s + 1.19i·19-s + 0.924i·23-s + 0.533·25-s − 0.286i·29-s − 1.44·31-s + (−0.225 − 1.21i)35-s − 0.727i·37-s − 0.0326i·41-s − 1.60·43-s + 0.669·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.430 - 0.902i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.430 - 0.902i)$
$L(1)$  $\approx$  $1.033441209$
$L(\frac12)$  $\approx$  $1.033441209$
$L(\frac{3}{2})$   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 \( 1 + (0.480 + 2.60i)T \)
good5 \( 1 - 2.76T + 5T^{2} \)
11 \( 1 + 5.75T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6.71iT - 17T^{2} \)
19 \( 1 - 5.20iT - 19T^{2} \)
23 \( 1 - 4.43iT - 23T^{2} \)
29 \( 1 + 1.54iT - 29T^{2} \)
31 \( 1 + 8.05T + 31T^{2} \)
37 \( 1 + 4.42iT - 37T^{2} \)
41 \( 1 + 0.209iT - 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 4.58T + 47T^{2} \)
53 \( 1 - 8.05iT - 53T^{2} \)
59 \( 1 - 5.53iT - 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 - 4.04T + 67T^{2} \)
71 \( 1 + 1.10iT - 71T^{2} \)
73 \( 1 + 9.59iT - 73T^{2} \)
79 \( 1 - 14.7iT - 79T^{2} \)
83 \( 1 + 8.86iT - 83T^{2} \)
89 \( 1 - 13.6iT - 89T^{2} \)
97 \( 1 + 4.58iT - 97T^{2} \)
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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

−8.629376904385313910025515893550, −7.84361777366613028594583737142, −7.37477127938161980279377202913, −6.20806995128437093257385318823, −5.82611836953688349254290369340, −5.13881480782538036290456628320, −3.98695242246424937911409404379, −3.32551604271545992745387387674, −2.09765569965617703909680316887, −1.42889940851340566984453041599, 0.26332106281570304303525063134, 1.90711247947262028754099243888, 2.60517567494007918950111362801, 3.17278347497016499469456710426, 4.92901647372317068830332305813, 5.12042339362848870934334143614, 5.89714548733922432877902197353, 6.65767312897578531330537454154, 7.41041024840888055307816951218, 8.396189662616573408137460068561

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