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Properties

Label 2-63e2-441.229-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.0284 - 0.999i$
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 + (0.988 + 0.149i)7-s + (−1.04 + 0.411i)13-s + (−0.900 + 0.433i)16-s + (1.35 + 0.781i)19-s + (−0.988 + 0.149i)25-s + (0.0747 + 0.997i)28-s + 1.36i·31-s + (−1.07 − 0.997i)37-s + (1.63 + 1.11i)43-s + (0.955 + 0.294i)49-s + (−0.634 − 0.930i)52-s + (0.574 + 0.131i)61-s + (−0.623 − 0.781i)64-s + 0.149·67-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)4-s + (0.988 + 0.149i)7-s + (−1.04 + 0.411i)13-s + (−0.900 + 0.433i)16-s + (1.35 + 0.781i)19-s + (−0.988 + 0.149i)25-s + (0.0747 + 0.997i)28-s + 1.36i·31-s + (−1.07 − 0.997i)37-s + (1.63 + 1.11i)43-s + (0.955 + 0.294i)49-s + (−0.634 − 0.930i)52-s + (0.574 + 0.131i)61-s + (−0.623 − 0.781i)64-s + 0.149·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0284 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0284 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.0284 - 0.999i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.0284 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.395045367\)
\(L(\frac12)\) \(\approx\) \(1.395045367\)
\(L(1)\) 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 \( 1 + (-0.988 - 0.149i)T \)
good2 \( 1 + (-0.222 - 0.974i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (-0.733 + 0.680i)T^{2} \)
13 \( 1 + (1.04 - 0.411i)T + (0.733 - 0.680i)T^{2} \)
17 \( 1 + (-0.826 + 0.563i)T^{2} \)
19 \( 1 + (-1.35 - 0.781i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (0.826 - 0.563i)T^{2} \)
31 \( 1 - 1.36iT - T^{2} \)
37 \( 1 + (1.07 + 0.997i)T + (0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.365 + 0.930i)T^{2} \)
43 \( 1 + (-1.63 - 1.11i)T + (0.365 + 0.930i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.0747 - 0.997i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.574 - 0.131i)T + (0.900 + 0.433i)T^{2} \)
67 \( 1 - 0.149T + T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.81 + 0.712i)T + (0.733 + 0.680i)T^{2} \)
79 \( 1 - 1.91T + T^{2} \)
83 \( 1 + (0.733 + 0.680i)T^{2} \)
89 \( 1 + (-0.955 + 0.294i)T^{2} \)
97 \( 1 + (-0.258 + 0.149i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.777993567817286078262029587844, −7.81776097240172895745958363240, −7.58906408473602283819383891239, −6.86383103832972743520780803844, −5.75463466139478873736157861714, −5.02568209564177940167078696068, −4.23020980976292041864229357557, −3.40192250227490158432254434138, −2.45791272737740348052753047611, −1.58206160332773536016319983584, 0.789482722549281147672072698049, 1.93668820033489476624710323548, 2.69560200876943562103179066782, 4.01070046757475289842651594233, 4.92879304240580244629773499794, 5.35378151567831336945279541731, 6.10419533491802846845484052422, 7.23113921813358744149202532372, 7.48484318619696480084867141011, 8.441502568991976030266837193866

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