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Properties

Label 2-63e2-441.326-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.943 - 0.331i$
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  + (0.623 + 0.781i)4-s + (0.826 − 0.563i)7-s + (−0.109 − 1.46i)13-s + (−0.222 + 0.974i)16-s + (0.900 + 1.56i)19-s + (0.826 + 0.563i)25-s + (0.955 + 0.294i)28-s − 1.97·31-s + (1.95 + 0.294i)37-s + (−1.21 − 1.12i)43-s + (0.365 − 0.930i)49-s + (1.07 − 0.997i)52-s + (0.455 − 0.571i)61-s + (−0.900 + 0.433i)64-s + 1.91·67-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)4-s + (0.826 − 0.563i)7-s + (−0.109 − 1.46i)13-s + (−0.222 + 0.974i)16-s + (0.900 + 1.56i)19-s + (0.826 + 0.563i)25-s + (0.955 + 0.294i)28-s − 1.97·31-s + (1.95 + 0.294i)37-s + (−1.21 − 1.12i)43-s + (0.365 − 0.930i)49-s + (1.07 − 0.997i)52-s + (0.455 − 0.571i)61-s + (−0.900 + 0.433i)64-s + 1.91·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.943 - 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.712605996\)
\(L(\frac12)\) \(\approx\) \(1.712605996\)
\(L(1)\) 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 \( 1 + (-0.826 + 0.563i)T \)
good2 \( 1 + (-0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.826 - 0.563i)T^{2} \)
11 \( 1 + (0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.109 + 1.46i)T + (-0.988 + 0.149i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.955 + 0.294i)T^{2} \)
29 \( 1 + (0.733 - 0.680i)T^{2} \)
31 \( 1 + 1.97T + T^{2} \)
37 \( 1 + (-1.95 - 0.294i)T + (0.955 + 0.294i)T^{2} \)
41 \( 1 + (-0.0747 + 0.997i)T^{2} \)
43 \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (-0.955 + 0.294i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (-0.455 + 0.571i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 - 1.91T + T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.0931 + 1.24i)T + (-0.988 - 0.149i)T^{2} \)
79 \( 1 - 0.730T + T^{2} \)
83 \( 1 + (0.988 + 0.149i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
97 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.338385337195517509033209102626, −7.85235227210671107697029624682, −7.46867354300416737173421873066, −6.61808219788132762450352515768, −5.60198721018634062475580778178, −5.05349151886264772729134475547, −3.78086364557959086151263729998, −3.41176196922833813286496515590, −2.28355826536084198621502043544, −1.24505604772280292092283920774, 1.18454278821053759701428743417, 2.13523697470875816880840198875, 2.81250637750135099199955592724, 4.21967753728484473788519381886, 4.97889113077939177761673598026, 5.52024303783440960318875643686, 6.51521970044423044261944261599, 6.99101005944351128465239985723, 7.75818706529117394426524195583, 8.740198488078288056953886618706

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