Properties

Degree $2$
Conductor $3969$
Sign $0.971 + 0.235i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 16-s + (−0.5 − 0.866i)25-s + (1 + 1.73i)37-s + (1 − 1.73i)43-s − 64-s + 2·67-s + 2·79-s + (0.5 + 0.866i)100-s + (1 − 1.73i)109-s + ⋯
L(s)  = 1  − 4-s + 16-s + (−0.5 − 0.866i)25-s + (1 + 1.73i)37-s + (1 − 1.73i)43-s − 64-s + 2·67-s + 2·79-s + (0.5 + 0.866i)100-s + (1 − 1.73i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.971 + 0.235i$
Motivic weight: \(0\)
Character: $\chi_{3969} (2971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.971 + 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9676890712\)
\(L(\frac12)\) \(\approx\) \(0.9676890712\)
\(L(1)\) 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 \( 1 + T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
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 \( 1 - T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 - 2T + T^{2} \)
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

−8.477871022431795773556003663855, −8.122830317066194019739345565571, −7.21374550682274048826263045610, −6.30882740193674643537076218907, −5.57306651088991432208630119969, −4.77721636568213698627598579167, −4.10985503883110089462179773535, −3.29365954359094964092295553037, −2.17288818483861411014490213707, −0.791363440723084137400474316082, 0.917085110099472635406967331121, 2.24873927156594603547048436836, 3.41014954310708957042753152505, 4.09363921636143933977865234583, 4.88573270404298101204509525849, 5.63008439754334195154618621950, 6.31611146052426385825735333878, 7.43366841098501318227454996021, 7.912239764178354488690315617026, 8.730356349045030904179477848138

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