Properties

Label 2-63e2-63.32-c0-0-6
Degree $2$
Conductor $3969$
Sign $-0.845 + 0.533i$
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 − 1.93i·11-s + (−0.133 + 0.232i)16-s + (−0.499 + 0.866i)22-s − 1.41i·23-s + 25-s + (−1.22 + 0.707i)29-s + (0.896 − 0.517i)32-s + (0.866 + 1.5i)37-s + (−0.866 − 1.5i)43-s + (−1.22 + 0.707i)44-s + (−0.366 + 0.633i)46-s + (−0.448 − 0.258i)50-s + ⋯
L(s)  = 1  + (−0.448 − 0.258i)2-s + (−0.366 − 0.633i)4-s + 0.896i·8-s − 1.93i·11-s + (−0.133 + 0.232i)16-s + (−0.499 + 0.866i)22-s − 1.41i·23-s + 25-s + (−1.22 + 0.707i)29-s + (0.896 − 0.517i)32-s + (0.866 + 1.5i)37-s + (−0.866 − 1.5i)43-s + (−1.22 + 0.707i)44-s + (−0.366 + 0.633i)46-s + (−0.448 − 0.258i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-0.845 + 0.533i$
Analytic conductor: \(1.98078\)
Root analytic conductor: \(1.40740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3969} (2321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3969,\ (\ :0),\ -0.845 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6653653586\)
\(L(\frac12)\) \(\approx\) \(0.6653653586\)
\(L(1)\) 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 \( 1 - T^{2} \)
11 \( 1 + 1.93iT - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.554646954230867738391888961867, −7.927001410987601618239050162700, −6.69021503946809839000162444174, −6.14349481176473752146328099442, −5.33970604590407859009142441907, −4.70294899942881768737566191399, −3.55198102112787491142878937702, −2.76127541702328020114893236623, −1.54187036197095550472126849503, −0.45984063823424267984120428279, 1.48920090802236468010164958472, 2.59655790297643723403200401274, 3.65846883298553042115121849468, 4.39528805680491112611635197698, 5.05751182772870397688502578275, 6.14606077210635931847613214876, 7.04909885095690567263490293352, 7.54792299048725992068210687139, 7.986818616569090869237486464507, 9.117680006782801880185431651798

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