Properties

Label 2-207-1.1-c5-0-36
Degree $2$
Conductor $207$
Sign $-1$
Analytic cond. $33.1994$
Root an. cond. $5.76189$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.04·2-s − 27.8·4-s − 42.3·5-s + 191.·7-s − 122.·8-s − 86.4·10-s + 655.·11-s − 932.·13-s + 391.·14-s + 641.·16-s + 344.·17-s − 548.·19-s + 1.17e3·20-s + 1.33e3·22-s + 529·23-s − 1.33e3·25-s − 1.90e3·26-s − 5.33e3·28-s − 4.39e3·29-s − 4.43e3·31-s + 5.21e3·32-s + 702.·34-s − 8.11e3·35-s − 1.14e4·37-s − 1.11e3·38-s + 5.17e3·40-s + 6.66e3·41-s + ⋯
L(s)  = 1  + 0.360·2-s − 0.869·4-s − 0.757·5-s + 1.47·7-s − 0.674·8-s − 0.273·10-s + 1.63·11-s − 1.53·13-s + 0.533·14-s + 0.626·16-s + 0.288·17-s − 0.348·19-s + 0.659·20-s + 0.589·22-s + 0.208·23-s − 0.425·25-s − 0.552·26-s − 1.28·28-s − 0.971·29-s − 0.828·31-s + 0.900·32-s + 0.104·34-s − 1.12·35-s − 1.37·37-s − 0.125·38-s + 0.511·40-s + 0.618·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(33.1994\)
Root analytic conductor: \(5.76189\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 207,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 - 529T \)
good2 \( 1 - 2.04T + 32T^{2} \)
5 \( 1 + 42.3T + 3.12e3T^{2} \)
7 \( 1 - 191.T + 1.68e4T^{2} \)
11 \( 1 - 655.T + 1.61e5T^{2} \)
13 \( 1 + 932.T + 3.71e5T^{2} \)
17 \( 1 - 344.T + 1.41e6T^{2} \)
19 \( 1 + 548.T + 2.47e6T^{2} \)
29 \( 1 + 4.39e3T + 2.05e7T^{2} \)
31 \( 1 + 4.43e3T + 2.86e7T^{2} \)
37 \( 1 + 1.14e4T + 6.93e7T^{2} \)
41 \( 1 - 6.66e3T + 1.15e8T^{2} \)
43 \( 1 + 2.28e4T + 1.47e8T^{2} \)
47 \( 1 + 1.79e4T + 2.29e8T^{2} \)
53 \( 1 + 4.56e3T + 4.18e8T^{2} \)
59 \( 1 - 2.32e4T + 7.14e8T^{2} \)
61 \( 1 - 2.51e4T + 8.44e8T^{2} \)
67 \( 1 + 6.21e3T + 1.35e9T^{2} \)
71 \( 1 + 6.80e3T + 1.80e9T^{2} \)
73 \( 1 + 3.29e4T + 2.07e9T^{2} \)
79 \( 1 + 2.10e4T + 3.07e9T^{2} \)
83 \( 1 + 2.83e4T + 3.93e9T^{2} \)
89 \( 1 + 1.37e5T + 5.58e9T^{2} \)
97 \( 1 + 3.22e4T + 8.58e9T^{2} \)
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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

−11.46339461494773579956165603883, −9.942262375578843200778828651903, −8.937244627866826920026453541290, −8.069789871867066558931947782610, −7.05371593967462817077313348123, −5.37835105955995206111827621247, −4.52245238586583310700619531510, −3.65337288200133782614945570384, −1.63564036329848118343167978314, 0, 1.63564036329848118343167978314, 3.65337288200133782614945570384, 4.52245238586583310700619531510, 5.37835105955995206111827621247, 7.05371593967462817077313348123, 8.069789871867066558931947782610, 8.937244627866826920026453541290, 9.942262375578843200778828651903, 11.46339461494773579956165603883

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