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Properties

Label 2-63e2-441.346-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.368 - 0.929i$
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 + (−0.365 + 0.930i)7-s + (0.167 + 1.11i)13-s + (0.826 − 0.563i)16-s + (1.35 − 0.781i)19-s + (0.623 − 0.781i)25-s + (0.0747 − 0.997i)28-s + (−1.17 + 0.680i)31-s + (−1.07 + 0.997i)37-s + (0.147 + 1.97i)43-s + (−0.733 − 0.680i)49-s + (−0.488 − 1.01i)52-s + (−0.173 + 0.563i)61-s + (−0.623 + 0.781i)64-s + (−0.0747 − 0.129i)67-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)4-s + (−0.365 + 0.930i)7-s + (0.167 + 1.11i)13-s + (0.826 − 0.563i)16-s + (1.35 − 0.781i)19-s + (0.623 − 0.781i)25-s + (0.0747 − 0.997i)28-s + (−1.17 + 0.680i)31-s + (−1.07 + 0.997i)37-s + (0.147 + 1.97i)43-s + (−0.733 − 0.680i)49-s + (−0.488 − 1.01i)52-s + (−0.173 + 0.563i)61-s + (−0.623 + 0.781i)64-s + (−0.0747 − 0.129i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.368 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7843007217\)
\(L(\frac12)\) \(\approx\) \(0.7843007217\)
\(L(1)\) 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 \( 1 + (0.955 - 0.294i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (-0.167 - 1.11i)T + (-0.955 + 0.294i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (-1.35 + 0.781i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.0747 - 0.997i)T^{2} \)
31 \( 1 + (1.17 - 0.680i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.07 - 0.997i)T + (0.0747 - 0.997i)T^{2} \)
41 \( 1 + (0.988 + 0.149i)T^{2} \)
43 \( 1 + (-0.147 - 1.97i)T + (-0.988 + 0.149i)T^{2} \)
47 \( 1 + (-0.955 + 0.294i)T^{2} \)
53 \( 1 + (0.0747 + 0.997i)T^{2} \)
59 \( 1 + (0.988 - 0.149i)T^{2} \)
61 \( 1 + (0.173 - 0.563i)T + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (1.81 - 0.712i)T + (0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.955 - 0.294i)T^{2} \)
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

−8.821440919486837036574882549647, −8.423544655168286723570486495339, −7.38255420733399915711455207613, −6.70988235977411229986595753534, −5.79414959744181509442700759133, −5.04184904230205882585813330333, −4.43589431695753973734189666490, −3.38961171421906460267116243551, −2.73415781699611075637741205391, −1.36411242775700653958271615671, 0.49910378846716486912532331841, 1.61488707150331169677926293600, 3.33356276553011146153534198468, 3.61610315987856279854662114898, 4.65240501456672409390431872915, 5.51266327822160087136090421902, 5.89063884801047376011873370998, 7.30348114594009785247241372793, 7.46282794213113685774996498449, 8.517057102574292304947683540542

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