Properties

Label 2-55506-1.1-c1-0-27
Degree $2$
Conductor $55506$
Sign $-1$
Analytic cond. $443.217$
Root an. cond. $21.0527$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 4·13-s − 3·14-s − 15-s + 16-s − 3·17-s − 18-s + 5·19-s + 20-s − 3·21-s + 22-s − 6·23-s + 24-s − 4·25-s − 4·26-s − 27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.10·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s − 4/5·25-s − 0.784·26-s − 0.192·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55506\)    =    \(2 \cdot 3 \cdot 11 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(443.217\)
Root analytic conductor: \(21.0527\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 55506,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 + T \)
29 \( 1 \)
good5 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.70697742971292, −14.05846941193080, −13.66451154257042, −13.24278361320674, −12.42915448670599, −11.97591311648672, −11.39723039710537, −11.05124223125146, −10.71616584048167, −9.976390984544785, −9.548096746673198, −9.040027807820999, −8.282873772786434, −7.939118656988666, −7.496693785791512, −6.722537127182395, −6.116660062908633, −5.732284858346842, −5.160640240002847, −4.426592776554973, −3.876986810164913, −2.997854218723034, −2.122340481947989, −1.627109686572717, −1.011069240588151, 0, 1.011069240588151, 1.627109686572717, 2.122340481947989, 2.997854218723034, 3.876986810164913, 4.426592776554973, 5.160640240002847, 5.732284858346842, 6.116660062908633, 6.722537127182395, 7.496693785791512, 7.939118656988666, 8.282873772786434, 9.040027807820999, 9.548096746673198, 9.976390984544785, 10.71616584048167, 11.05124223125146, 11.39723039710537, 11.97591311648672, 12.42915448670599, 13.24278361320674, 13.66451154257042, 14.05846941193080, 14.70697742971292

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