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Properties

Label 2-63e2-63.61-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.805 + 0.592i$
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  + (−0.866 − 1.5i)2-s + (−1 + 1.73i)4-s + 1.73·8-s − 1.73·11-s + (−0.5 − 0.866i)16-s + (1.49 + 2.59i)22-s + 25-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s + (1.73 − 3i)44-s + (−0.866 − 1.5i)50-s + (0.866 + 1.5i)53-s − 0.999·64-s + (0.5 − 0.866i)67-s + 1.73·71-s + ⋯
L(s)  = 1  + (−0.866 − 1.5i)2-s + (−1 + 1.73i)4-s + 1.73·8-s − 1.73·11-s + (−0.5 − 0.866i)16-s + (1.49 + 2.59i)22-s + 25-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)43-s + (1.73 − 3i)44-s + (−0.866 − 1.5i)50-s + (0.866 + 1.5i)53-s − 0.999·64-s + (0.5 − 0.866i)67-s + 1.73·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.805 + 0.592i$
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: \((2,\ 3969,\ (\ :0),\ 0.805 + 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5405542964\)
\(L(\frac12)\) \(\approx\) \(0.5405542964\)
\(L(1)\) 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 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + 1.73T + T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.671327501493717784281036566134, −8.091516153824036048385399786082, −7.51476779002436115469532332340, −6.46688126773626174252899360295, −5.31606543546773728417212005199, −4.64613576053812417333478091661, −3.54191110121503720780547574156, −2.81592891573802970161485433766, −2.17209944814564299118856601607, −0.961905073134386469154891176553, 0.50854380235820461590738943848, 2.11559349590874141738305129526, 3.26124633349112186740704832729, 4.59785516865929703006415627313, 5.35975289850703365644057180774, 5.72088573612729399074510549131, 6.89371454749896018709021900628, 7.12423433365009865025572122873, 8.151428713411321259889557371959, 8.331937942706490276154665600126

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