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Properties

Label 2-63e2-441.124-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.772 - 0.634i$
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  + (−0.365 − 0.930i)4-s + (−0.0747 + 0.997i)7-s + (−0.766 + 1.12i)13-s + (−0.733 + 0.680i)16-s + (−0.751 − 0.433i)19-s + (−0.900 − 0.433i)25-s + (0.955 − 0.294i)28-s + (0.258 + 0.149i)31-s + (−1.95 + 0.294i)37-s + (−1.57 − 0.487i)43-s + (−0.988 − 0.149i)49-s + (1.32 + 0.302i)52-s + (−1.73 − 0.680i)61-s + (0.900 + 0.433i)64-s + (−0.955 + 1.65i)67-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)4-s + (−0.0747 + 0.997i)7-s + (−0.766 + 1.12i)13-s + (−0.733 + 0.680i)16-s + (−0.751 − 0.433i)19-s + (−0.900 − 0.433i)25-s + (0.955 − 0.294i)28-s + (0.258 + 0.149i)31-s + (−1.95 + 0.294i)37-s + (−1.57 − 0.487i)43-s + (−0.988 − 0.149i)49-s + (1.32 + 0.302i)52-s + (−1.73 − 0.680i)61-s + (0.900 + 0.433i)64-s + (−0.955 + 1.65i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.772 - 0.634i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1594, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.772 - 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2727212436\)
\(L(\frac12)\) \(\approx\) \(0.2727212436\)
\(L(1)\) 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 \( 1 + (0.0747 - 0.997i)T \)
good2 \( 1 + (0.365 + 0.930i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (0.766 - 1.12i)T + (-0.365 - 0.930i)T^{2} \)
17 \( 1 + (0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.751 + 0.433i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (0.955 - 0.294i)T^{2} \)
31 \( 1 + (-0.258 - 0.149i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.95 - 0.294i)T + (0.955 - 0.294i)T^{2} \)
41 \( 1 + (-0.826 + 0.563i)T^{2} \)
43 \( 1 + (1.57 + 0.487i)T + (0.826 + 0.563i)T^{2} \)
47 \( 1 + (-0.365 - 0.930i)T^{2} \)
53 \( 1 + (0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (1.73 + 0.680i)T + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-1.55 + 0.116i)T + (0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.365 + 0.632i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.365 + 0.930i)T^{2} \)
89 \( 1 + (-0.365 + 0.930i)T^{2} \)
97 \( 1 + (-0.975 - 0.563i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.838590470246098502669794011072, −8.554969646127169117557102556429, −7.34970458971715457620571433426, −6.51702807394104356457390505223, −6.05349728393936848659621120562, −5.03672113677345678144961668116, −4.70262773830148616302779971882, −3.58484200205995139102355500421, −2.30767757282308522270682861132, −1.73367153749849537062125931945, 0.14215645742292930269964619854, 1.83805941881446223057329949175, 3.09775382930557823863991953268, 3.61998346485067711844705851138, 4.50800879976893349563074696722, 5.17040075117623616752766003238, 6.23637717363177241884301148002, 7.11272735513857426189724806597, 7.67606923027040132982870027325, 8.183395778473548497160757121907

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