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Properties

Label 2-63e2-21.11-c0-0-0
Degree $2$
Conductor $3969$
Sign $0.386 - 0.922i$
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  + (−0.448 + 0.258i)2-s + (−0.366 + 0.633i)4-s − 0.896i·8-s + (−1.67 − 0.965i)11-s + (−0.133 − 0.232i)16-s + 22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + 1.41i·29-s + (0.896 + 0.517i)32-s + (0.866 + 1.5i)37-s + 1.73·43-s + (1.22 − 0.707i)44-s + (−0.366 + 0.633i)46-s − 0.517i·50-s + ⋯
L(s)  = 1  + (−0.448 + 0.258i)2-s + (−0.366 + 0.633i)4-s − 0.896i·8-s + (−1.67 − 0.965i)11-s + (−0.133 − 0.232i)16-s + 22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + 1.41i·29-s + (0.896 + 0.517i)32-s + (0.866 + 1.5i)37-s + 1.73·43-s + (1.22 − 0.707i)44-s + (−0.366 + 0.633i)46-s − 0.517i·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (3644, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7308416681\)
\(L(\frac12)\) \(\approx\) \(0.7308416681\)
\(L(1)\) 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 \( 1 \)
good2 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.717930127872040356929781881458, −8.059435998891845262528874226883, −7.47239778991555633130339474854, −6.81288515130580458099642169797, −5.76030891185126050509117211030, −5.09841663855521756040666959584, −4.22378759215897192505886178964, −3.15567588365237552681398218796, −2.69517042453187089795009130359, −0.932468951680064599897793410216, 0.63043382077380948665986072879, 2.10892775191797915234397979703, 2.58851018559407380556433287576, 4.04511700137861091433301372346, 4.80915136599830504029786775983, 5.48793825152450862609782500853, 6.11054581579201717084293667679, 7.33913280722456109669426729687, 7.75110255539701597411317049921, 8.570328646383324431889338208740

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