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Properties

Label 2-63e2-63.23-c0-0-3
Degree $2$
Conductor $3969$
Sign $0.212 - 0.977i$
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  + 0.517i·2-s + 0.732·4-s + 0.896i·8-s + (1.67 + 0.965i)11-s + 0.267·16-s + (−0.499 + 0.866i)22-s + (−1.22 + 0.707i)23-s + (−0.5 + 0.866i)25-s + (−1.22 + 0.707i)29-s + 1.03i·32-s + (0.866 − 1.5i)37-s + (−0.866 − 1.5i)43-s + (1.22 + 0.707i)44-s + (−0.366 − 0.633i)46-s + (−0.448 − 0.258i)50-s + ⋯
L(s)  = 1  + 0.517i·2-s + 0.732·4-s + 0.896i·8-s + (1.67 + 0.965i)11-s + 0.267·16-s + (−0.499 + 0.866i)22-s + (−1.22 + 0.707i)23-s + (−0.5 + 0.866i)25-s + (−1.22 + 0.707i)29-s + 1.03i·32-s + (0.866 − 1.5i)37-s + (−0.866 − 1.5i)43-s + (1.22 + 0.707i)44-s + (−0.366 − 0.633i)46-s + (−0.448 − 0.258i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.212 - 0.977i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.212 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.718946179\)
\(L(\frac12)\) \(\approx\) \(1.718946179\)
\(L(1)\) 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 \( 1 - 0.517iT - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - 0.517iT - T^{2} \)
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

−8.763201007103532861519578658603, −7.85550411076960168935146639368, −7.11850399306535039418618241681, −6.84778134025129931766335996753, −5.82272223587531159417701303174, −5.38420301548746087082580783927, −4.08455498119408133648635132028, −3.60083590036671335199397803247, −2.15557743010744772309195651888, −1.60638479577695219851234057612, 1.00831463843115066380729500930, 1.98219356838875380306683478264, 2.93009095769573803338724445198, 3.82738903523734381517145936458, 4.34138498793827936382562673728, 5.82108018273530613727131414236, 6.26541377664442689998127862965, 6.82103479930528317144680722036, 7.85931395091329925684299625098, 8.416890235224350640480494581633

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