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Properties

Label 2-63e2-441.202-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.851 + 0.524i$
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  + (−0.955 − 0.294i)4-s + (0.988 + 0.149i)7-s + (−0.807 + 0.317i)13-s + (0.826 + 0.563i)16-s − 1.56i·19-s + (0.365 − 0.930i)25-s + (−0.900 − 0.433i)28-s + (1.68 + 0.974i)31-s + (−0.0990 + 0.433i)37-s + (−1.03 − 0.702i)43-s + (0.955 + 0.294i)49-s + (0.865 − 0.0648i)52-s + (0.574 + 1.86i)61-s + (−0.623 − 0.781i)64-s + (0.900 − 1.56i)67-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)4-s + (0.988 + 0.149i)7-s + (−0.807 + 0.317i)13-s + (0.826 + 0.563i)16-s − 1.56i·19-s + (0.365 − 0.930i)25-s + (−0.900 − 0.433i)28-s + (1.68 + 0.974i)31-s + (−0.0990 + 0.433i)37-s + (−1.03 − 0.702i)43-s + (0.955 + 0.294i)49-s + (0.865 − 0.0648i)52-s + (0.574 + 1.86i)61-s + (−0.623 − 0.781i)64-s + (0.900 − 1.56i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.851 + 0.524i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.851 + 0.524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.101484707\)
\(L(\frac12)\) \(\approx\) \(1.101484707\)
\(L(1)\) 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 \( 1 + (0.955 + 0.294i)T^{2} \)
5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (0.955 + 0.294i)T^{2} \)
13 \( 1 + (0.807 - 0.317i)T + (0.733 - 0.680i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 + 1.56iT - T^{2} \)
23 \( 1 + (0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.826 - 0.563i)T^{2} \)
31 \( 1 + (-1.68 - 0.974i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.365 + 0.930i)T^{2} \)
43 \( 1 + (1.03 + 0.702i)T + (0.365 + 0.930i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (-0.574 - 1.86i)T + (-0.826 + 0.563i)T^{2} \)
67 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.52 + 1.21i)T + (0.222 - 0.974i)T^{2} \)
79 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.733 + 0.680i)T^{2} \)
89 \( 1 + (0.222 - 0.974i)T^{2} \)
97 \( 1 + (-1.35 + 0.781i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.571955802467856500000883712662, −8.075926405300646811948925021283, −7.09499321288870416689066719117, −6.40092923474097253184699400159, −5.27513317330495522819739164818, −4.79911477252678391753452617301, −4.33967912161438184105776761513, −3.06782797958846252432768834904, −2.07293944117506511837533828803, −0.815230361614382745936187018131, 1.06466994704824577016372702640, 2.27818915507025931502229666939, 3.44855446804311332241391351502, 4.16841357612053163728846420536, 4.99322608466325288399806635943, 5.43644205077591242887161273121, 6.49432673238754541954835039247, 7.55771202458326962922659107924, 8.031549056498937292990219501083, 8.476787605909193751334595644486

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