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Properties

Label 2-63e2-21.11-c0-0-2
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  + (−1.67 + 0.965i)2-s + (1.36 − 2.36i)4-s + 3.34i·8-s + (−0.448 − 0.258i)11-s + (−1.86 − 3.23i)16-s + 22-s + (−1.22 + 0.707i)23-s + (−0.5 + 0.866i)25-s − 1.41i·29-s + (3.34 + 1.93i)32-s + (−0.866 − 1.5i)37-s − 1.73·43-s + (−1.22 + 0.707i)44-s + (1.36 − 2.36i)46-s − 1.93i·50-s + ⋯
L(s)  = 1  + (−1.67 + 0.965i)2-s + (1.36 − 2.36i)4-s + 3.34i·8-s + (−0.448 − 0.258i)11-s + (−1.86 − 3.23i)16-s + 22-s + (−1.22 + 0.707i)23-s + (−0.5 + 0.866i)25-s − 1.41i·29-s + (3.34 + 1.93i)32-s + (−0.866 − 1.5i)37-s − 1.73·43-s + (−1.22 + 0.707i)44-s + (1.36 − 2.36i)46-s − 1.93i·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

\(L(\frac{1}{2})\) \(\approx\) \(0.09072407784\)
\(L(\frac12)\) \(\approx\) \(0.09072407784\)
\(L(1)\) 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 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.93iT - T^{2} \)
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

−8.301336406056065361341149252434, −7.76064975306524694855082479655, −7.23596615780375433203156049682, −6.34549631809883904613657453902, −5.74812380169966029068050294523, −5.11815120429724263147208474623, −3.76652678241752782786184525518, −2.37712105266491400171346872291, −1.55196480276948336071870221606, −0.087514322400614007152808824734, 1.42010340399138272867920802444, 2.24549232814529287851216095157, 3.09082773534407776037074698998, 3.92349057800725349483473672611, 4.99014564983376236837765111272, 6.33190515267157509450974800745, 6.93196172945283703034475296996, 7.75533312253702936992970141360, 8.379956180925250933883671147215, 8.765220695549731249824343273881

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