Properties

Degree $2$
Conductor $3969$
Sign $0.400 + 0.916i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.499 − 0.866i)16-s + 25-s + (1 − 1.73i)37-s + (1 − 1.73i)43-s − 0.999·64-s + (−1 + 1.73i)67-s + (−1 − 1.73i)79-s + (0.5 − 0.866i)100-s + (1 + 1.73i)109-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.499 − 0.866i)16-s + 25-s + (1 − 1.73i)37-s + (1 − 1.73i)43-s − 0.999·64-s + (−1 + 1.73i)67-s + (−1 − 1.73i)79-s + (0.5 − 0.866i)100-s + (1 + 1.73i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.400 + 0.916i$
Motivic weight: \(0\)
Character: $\chi_{3969} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.444516188\)
\(L(\frac12)\) \(\approx\) \(1.444516188\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.696551878464449260899036447508, −7.49425000426323712467122378060, −7.13089756279581957694855461432, −6.16488249538535962624134670026, −5.65774059596497633754107298043, −4.83915600892579052270229787472, −3.96314207153873087304259446114, −2.81609312274742620477826042413, −2.02701488115012690640643308703, −0.861532186567199083432273547285, 1.38842073496557968953249765669, 2.62912072414162170463541512776, 3.16377550659673684330343252930, 4.20832186045896750308016583864, 4.86391438647474554353710761076, 6.03162458275508097594314718707, 6.59395707194047581386396098959, 7.37593136056098576442766722920, 8.013244572062605954710680295602, 8.611994860749644409645851692669

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