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Properties

Label 2-63e2-441.176-c0-0-0
Degree $2$
Conductor $3969$
Sign $-0.935 - 0.352i$
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  + (−0.988 + 0.149i)4-s + (0.0747 + 0.997i)7-s + (−0.367 + 0.250i)13-s + (0.955 − 0.294i)16-s − 1.80·19-s + (0.826 + 0.563i)25-s + (−0.222 − 0.974i)28-s + (−0.623 + 1.07i)31-s + (0.777 − 0.974i)37-s + (−1.72 + 0.531i)43-s + (−0.988 + 0.149i)49-s + (0.326 − 0.302i)52-s + (−1.23 − 0.185i)61-s + (−0.900 + 0.433i)64-s + (0.222 − 0.385i)67-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)4-s + (0.0747 + 0.997i)7-s + (−0.367 + 0.250i)13-s + (0.955 − 0.294i)16-s − 1.80·19-s + (0.826 + 0.563i)25-s + (−0.222 − 0.974i)28-s + (−0.623 + 1.07i)31-s + (0.777 − 0.974i)37-s + (−1.72 + 0.531i)43-s + (−0.988 + 0.149i)49-s + (0.326 − 0.302i)52-s + (−1.23 − 0.185i)61-s + (−0.900 + 0.433i)64-s + (0.222 − 0.385i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2969, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4033636226\)
\(L(\frac12)\) \(\approx\) \(0.4033636226\)
\(L(1)\) 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 \( 1 + (0.988 - 0.149i)T^{2} \)
5 \( 1 + (-0.826 - 0.563i)T^{2} \)
11 \( 1 + (0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.367 - 0.250i)T + (0.365 - 0.930i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 + (-0.955 + 0.294i)T^{2} \)
29 \( 1 + (-0.955 - 0.294i)T^{2} \)
31 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
41 \( 1 + (-0.826 - 0.563i)T^{2} \)
43 \( 1 + (1.72 - 0.531i)T + (0.826 - 0.563i)T^{2} \)
47 \( 1 + (0.988 - 0.149i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (1.23 + 0.185i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.365 - 0.930i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.895993556661981979170913124957, −8.455638054098831206457380505766, −7.65458331181708348341082535631, −6.68693163155314851147793674067, −5.94037007886548607782835055725, −5.07974038887916233346782785444, −4.58025333599096028745708904288, −3.61926708428809899313869952191, −2.69937923216900927729842354529, −1.63090978202194504542878852035, 0.23324496212463758812176160497, 1.55575303614403566569528683795, 2.86598365256134051240901778097, 3.95495543023246849060410008026, 4.42704562214122121404681034876, 5.13588951212088888743627809040, 6.13875684758807246011159951866, 6.80683190051485750549519013727, 7.74183871226026760357186940811, 8.327040611102826874079157392570

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