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Properties

Label 2-63e2-63.40-c0-0-5
Degree $2$
Conductor $3969$
Sign $0.959 + 0.281i$
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  + 1.73·2-s + 1.99·4-s + 1.73·8-s + (0.866 − 1.5i)11-s + 0.999·16-s + (1.49 − 2.59i)22-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)43-s + (1.73 − 2.99i)44-s + (−0.866 + 1.49i)50-s + (0.866 + 1.5i)53-s − 1.00·64-s − 67-s + 1.73·71-s + ⋯
L(s)  = 1  + 1.73·2-s + 1.99·4-s + 1.73·8-s + (0.866 − 1.5i)11-s + 0.999·16-s + (1.49 − 2.59i)22-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)43-s + (1.73 − 2.99i)44-s + (−0.866 + 1.49i)50-s + (0.866 + 1.5i)53-s − 1.00·64-s − 67-s + 1.73·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.959 + 0.281i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.959 + 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.710714813\)
\(L(\frac12)\) \(\approx\) \(3.710714813\)
\(L(1)\) 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 \( 1 - 1.73T + T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
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

−8.581076974222943047044791257425, −7.58973058906621663075698849200, −6.79860955629731887683509304928, −6.14687332226755995032182462644, −5.59932681794536637332640113825, −4.86303142459192649329745923612, −3.87145888329129816713328400606, −3.47279728305178614986613421675, −2.58958799035717831687024240807, −1.38517412982404466569216303836, 1.71919932899013022003142714630, 2.43775079389472226373642379069, 3.52816982412546769549423176406, 4.19565742368966711099103853509, 4.76157698582531359048443328679, 5.54566697592830877382588787753, 6.37724201018458806345156080006, 6.90982894794133373496527069335, 7.55025245273937725771312558078, 8.572041131191569257374867168657

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