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Properties

Label 2-63e2-441.11-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.484 - 0.874i$
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  + (−0.900 + 0.433i)4-s + (0.955 + 0.294i)7-s + (−0.535 + 0.496i)13-s + (0.623 − 0.781i)16-s + (0.222 + 0.385i)19-s + (0.955 − 0.294i)25-s + (−0.988 + 0.149i)28-s + 0.149·31-s + (0.0111 + 0.149i)37-s + (0.698 + 1.77i)43-s + (0.826 + 0.563i)49-s + (0.266 − 0.680i)52-s + (−1.48 − 0.716i)61-s + (−0.222 + 0.974i)64-s − 1.97·67-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)4-s + (0.955 + 0.294i)7-s + (−0.535 + 0.496i)13-s + (0.623 − 0.781i)16-s + (0.222 + 0.385i)19-s + (0.955 − 0.294i)25-s + (−0.988 + 0.149i)28-s + 0.149·31-s + (0.0111 + 0.149i)37-s + (0.698 + 1.77i)43-s + (0.826 + 0.563i)49-s + (0.266 − 0.680i)52-s + (−1.48 − 0.716i)61-s + (−0.222 + 0.974i)64-s − 1.97·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.484 - 0.874i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.484 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102838134\)
\(L(\frac12)\) \(\approx\) \(1.102838134\)
\(L(1)\) 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 \( 1 + (-0.955 - 0.294i)T \)
good2 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (-0.955 + 0.294i)T^{2} \)
11 \( 1 + (-0.0747 + 0.997i)T^{2} \)
13 \( 1 + (0.535 - 0.496i)T + (0.0747 - 0.997i)T^{2} \)
17 \( 1 + (-0.365 + 0.930i)T^{2} \)
19 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.988 + 0.149i)T^{2} \)
29 \( 1 + (-0.365 + 0.930i)T^{2} \)
31 \( 1 - 0.149T + T^{2} \)
37 \( 1 + (-0.0111 - 0.149i)T + (-0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.733 + 0.680i)T^{2} \)
43 \( 1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 + 1.97T + T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.32 - 1.22i)T + (0.0747 + 0.997i)T^{2} \)
79 \( 1 - 1.65T + T^{2} \)
83 \( 1 + (-0.0747 - 0.997i)T^{2} \)
89 \( 1 + (-0.826 + 0.563i)T^{2} \)
97 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.794649467028619166412179128475, −7.967959203072811096052619520051, −7.61200146025109537738038575220, −6.57277073790359494980595835304, −5.63943393089384408850796743501, −4.76891822016175113730270104042, −4.47121629312509903095846624973, −3.37964173414762376775485296842, −2.43791601987764839432208632350, −1.20787875589515877330683899581, 0.75232140979003004764444373996, 1.86944991938733577122759715191, 3.09300283747516644534754523664, 4.11642944881341256670580767626, 4.83323256633405762880660304771, 5.31201545536126222880877260232, 6.15774171374552983681031488775, 7.26114800275356004172969173430, 7.75092517544940234895021737577, 8.681650112581681099623643001359

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