Properties

Label 2-63e2-441.32-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 + (−1.21 + 0.825i)13-s + (−0.733 − 0.680i)16-s + (0.900 + 1.56i)19-s + (−0.900 + 0.433i)25-s + (0.955 + 0.294i)28-s + (0.988 + 1.71i)31-s + (1.95 + 0.294i)37-s + (1.57 − 0.487i)43-s + (−0.988 + 0.149i)49-s + (0.326 + 1.42i)52-s + (0.266 + 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 + (−1.21 + 0.825i)13-s + (−0.733 − 0.680i)16-s + (0.900 + 1.56i)19-s + (−0.900 + 0.433i)25-s + (0.955 + 0.294i)28-s + (0.988 + 1.71i)31-s + (1.95 + 0.294i)37-s + (1.57 − 0.487i)43-s + (−0.988 + 0.149i)49-s + (0.326 + 1.42i)52-s + (0.266 + 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: $\chi_{3969} (620, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.772 - 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.276972266\)
\(L(\frac12)\) \(\approx\) \(1.276972266\)
\(L(1)\) 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 \( 1 + (1.21 - 0.825i)T + (0.365 - 0.930i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.955 - 0.294i)T^{2} \)
31 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.266 - 0.680i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.0931 + 1.24i)T + (-0.988 - 0.149i)T^{2} \)
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 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.866790545606726538354695494104, −7.87563068078563777556449723166, −7.27122982954700024643430594507, −6.31927180229590569101573332811, −5.80148160802590582683154752334, −5.10048812285301175282612111486, −4.35112289285683546176733425848, −3.05057986894209817843924927664, −2.22002814205859726548298908533, −1.38987054788014594785333362989, 0.73268479139246417533319110664, 2.47606882818954749292592667562, 2.86765774239797350607477306055, 4.11395926803217419622627675793, 4.49317589647151411060465665340, 5.60615945717965102058107955360, 6.52672098879739888255493376093, 7.38948991360448276630071769278, 7.63514906749039895972721846514, 8.268818009946492256621038665983

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