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Properties

Label 2-539-11.10-c0-0-2
Degree $2$
Conductor $539$
Sign $1$
Analytic cond. $0.268996$
Root an. cond. $0.518648$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 4-s − 1.41·5-s + 1.00·9-s − 11-s + 1.41·12-s − 2.00·15-s + 16-s − 1.41·20-s + 1.00·25-s − 1.41·31-s − 1.41·33-s + 1.00·36-s − 2·37-s − 44-s − 1.41·45-s + 1.41·47-s + 1.41·48-s + 1.41·55-s + 1.41·59-s − 2.00·60-s + 64-s + 1.41·75-s − 1.41·80-s − 0.999·81-s + 1.41·89-s − 2.00·93-s + ⋯
L(s)  = 1  + 1.41·3-s + 4-s − 1.41·5-s + 1.00·9-s − 11-s + 1.41·12-s − 2.00·15-s + 16-s − 1.41·20-s + 1.00·25-s − 1.41·31-s − 1.41·33-s + 1.00·36-s − 2·37-s − 44-s − 1.41·45-s + 1.41·47-s + 1.41·48-s + 1.41·55-s + 1.41·59-s − 2.00·60-s + 64-s + 1.41·75-s − 1.41·80-s − 0.999·81-s + 1.41·89-s − 2.00·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.268996\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (197, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.284049929\)
\(L(\frac12)\) \(\approx\) \(1.284049929\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 + T \)
good2 \( 1 - T^{2} \)
3 \( 1 - 1.41T + T^{2} \)
5 \( 1 + 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−10.99109869371810235694453129510, −10.28028037251261176669156014481, −8.992451751449390382892384515393, −8.200262039299082806198192163323, −7.57426434045455978583978161806, −7.00599498043269359551063041368, −5.40729611683183440913031946823, −3.89061318589137670208299091502, −3.18608592665119827727950116462, −2.13715644622010918740527976330, 2.13715644622010918740527976330, 3.18608592665119827727950116462, 3.89061318589137670208299091502, 5.40729611683183440913031946823, 7.00599498043269359551063041368, 7.57426434045455978583978161806, 8.200262039299082806198192163323, 8.992451751449390382892384515393, 10.28028037251261176669156014481, 10.99109869371810235694453129510

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