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Properties

Label 2-63e2-63.61-c0-0-5
Degree $2$
Conductor $3969$
Sign $0.400 + 0.916i$
Analytic cond. $1.98078$
Root an. cond. $1.40740$
Motivic weight $0$
Arithmetic yes
Rational no
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 + (1.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 25-s + (−1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s − 1.73i·52-s + (1.5 − 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + (1.5 + 0.866i)97-s + (0.5 − 0.866i)100-s − 1.73i·103-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (1.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 25-s + (−1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s − 1.73i·52-s + (1.5 − 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + (1.5 + 0.866i)97-s + (0.5 − 0.866i)100-s − 1.73i·103-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: $0.400 + 0.916i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.535993797\)
\(L(\frac12)\) \(\approx\) \(1.535993797\)
\(L(1)\) 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 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.488633838646607235813003965186, −7.80075252924838030355791552784, −6.84981517408924346700867204470, −6.29974580381200253090656770581, −5.56371046908188918456572519489, −4.96477954992769936883061203332, −3.79591428457002125219018750350, −3.00137710005122716160463668300, −1.88536883880065982094522310803, −0.929211114693954615932929221806, 1.46707539226908886210842240472, 2.40538823173229163531696321226, 3.55601717595189937183127828890, 3.86665799230948920283244248292, 5.01392596827841682014063889009, 5.93867407403977119652919655640, 6.77366855644605466750994712019, 7.15261515896159011067601935442, 8.093556139875731936879771478104, 8.873187281100141177641773149947

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