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Properties

Label 2-63e2-441.137-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.620 + 0.784i$
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  + (−0.222 − 0.974i)4-s + (0.365 + 0.930i)7-s + (−0.147 − 0.0222i)13-s + (−0.900 + 0.433i)16-s + (−0.623 − 1.07i)19-s + (0.365 − 0.930i)25-s + (0.826 − 0.563i)28-s + 1.91·31-s + (1.82 − 0.563i)37-s + (0.0546 − 0.728i)43-s + (−0.733 + 0.680i)49-s + (0.0111 + 0.149i)52-s + (0.326 − 1.42i)61-s + (0.623 + 0.781i)64-s + 1.65·67-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)4-s + (0.365 + 0.930i)7-s + (−0.147 − 0.0222i)13-s + (−0.900 + 0.433i)16-s + (−0.623 − 1.07i)19-s + (0.365 − 0.930i)25-s + (0.826 − 0.563i)28-s + 1.91·31-s + (1.82 − 0.563i)37-s + (0.0546 − 0.728i)43-s + (−0.733 + 0.680i)49-s + (0.0111 + 0.149i)52-s + (0.326 − 1.42i)61-s + (0.623 + 0.781i)64-s + 1.65·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.620 + 0.784i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.620 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229062532\)
\(L(\frac12)\) \(\approx\) \(1.229062532\)
\(L(1)\) 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 \( 1 + (-0.365 - 0.930i)T \)
good2 \( 1 + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (-0.955 - 0.294i)T^{2} \)
13 \( 1 + (0.147 + 0.0222i)T + (0.955 + 0.294i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.826 - 0.563i)T^{2} \)
29 \( 1 + (-0.0747 - 0.997i)T^{2} \)
31 \( 1 - 1.91T + T^{2} \)
37 \( 1 + (-1.82 + 0.563i)T + (0.826 - 0.563i)T^{2} \)
41 \( 1 + (0.988 - 0.149i)T^{2} \)
43 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.326 + 1.42i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 - 1.65T + T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.440 + 0.0663i)T + (0.955 - 0.294i)T^{2} \)
79 \( 1 + 1.46T + T^{2} \)
83 \( 1 + (-0.955 + 0.294i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.530963865060158477878229860343, −8.043972948293588329556570552849, −6.79607684556081500833912335274, −6.31457320729128406740797941250, −5.53298613910437817711946788500, −4.80362855530982394143229998384, −4.24825159872152711848477542597, −2.72343705761253584003044028453, −2.15929147613364424943647711626, −0.815110193680977345320228354288, 1.17508361479285033514065077375, 2.51063234929655359585840749181, 3.38371346208140613294558013328, 4.25971430301378353105163066386, 4.64712903492896203555838896352, 5.82623036668321068750488082369, 6.73423423219675674730786226621, 7.34444318447254337631399682334, 8.207834795118127400554312231086, 8.322269144153032845223979541079

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