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Properties

Label 2-63e2-63.23-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.458 + 0.888i$
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  + 1.41i·2-s − 1.00·4-s + (−1.22 − 0.707i)11-s − 0.999·16-s + (1.00 − 1.73i)22-s + (−1.22 + 0.707i)23-s + (−0.5 + 0.866i)25-s + (−1.22 + 0.707i)29-s − 1.41i·32-s + (1.22 + 0.707i)44-s + (−1.00 − 1.73i)46-s + (−1.22 − 0.707i)50-s + (−1.22 + 0.707i)53-s + (−1.00 − 1.73i)58-s + 1.00·64-s + ⋯
L(s)  = 1  + 1.41i·2-s − 1.00·4-s + (−1.22 − 0.707i)11-s − 0.999·16-s + (1.00 − 1.73i)22-s + (−1.22 + 0.707i)23-s + (−0.5 + 0.866i)25-s + (−1.22 + 0.707i)29-s − 1.41i·32-s + (1.22 + 0.707i)44-s + (−1.00 − 1.73i)46-s + (−1.22 − 0.707i)50-s + (−1.22 + 0.707i)53-s + (−1.00 − 1.73i)58-s + 1.00·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.458 + 0.888i$
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: \((2,\ 3969,\ (\ :0),\ -0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4326781264\)
\(L(\frac12)\) \(\approx\) \(0.4326781264\)
\(L(1)\) 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 \( 1 - 1.41iT - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
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

−8.927651816768432355403037236950, −7.997121144156382795768892037610, −7.75297565279814815606334149155, −7.03062081436063520609423126865, −6.06602691552107228512197709700, −5.61571589940076969099699448195, −5.02442634269376999756962356618, −3.97426676866733277205237271218, −2.99499326994217563196980736520, −1.83859264676624976910908626637, 0.21760256341572365171808533235, 1.86013173053837737019475651569, 2.34734539959616413010260208756, 3.28656304363392551568212806775, 4.19051879121676900180348697477, 4.78594537069317133141177355301, 5.81624480162639099685164738764, 6.63732377034650138063010920818, 7.67722249811462792636593517149, 8.121535911076256442523573473241

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