Properties

Label 2-207-1.1-c5-0-44
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  + 9.32·2-s + 54.9·4-s − 99.7·5-s + 125.·7-s + 214.·8-s − 930.·10-s − 177.·11-s − 919.·13-s + 1.17e3·14-s + 240.·16-s − 1.56e3·17-s + 447.·19-s − 5.48e3·20-s − 1.65e3·22-s + 529·23-s + 6.81e3·25-s − 8.57e3·26-s + 6.92e3·28-s + 1.29e3·29-s − 6.65e3·31-s − 4.61e3·32-s − 1.46e4·34-s − 1.25e4·35-s − 8.05e3·37-s + 4.17e3·38-s − 2.13e4·40-s − 1.79e4·41-s + ⋯
L(s)  = 1  + 1.64·2-s + 1.71·4-s − 1.78·5-s + 0.971·7-s + 1.18·8-s − 2.94·10-s − 0.442·11-s − 1.50·13-s + 1.60·14-s + 0.235·16-s − 1.31·17-s + 0.284·19-s − 3.06·20-s − 0.730·22-s + 0.208·23-s + 2.18·25-s − 2.48·26-s + 1.67·28-s + 0.286·29-s − 1.24·31-s − 0.797·32-s − 2.17·34-s − 1.73·35-s − 0.967·37-s + 0.469·38-s − 2.11·40-s − 1.66·41-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: $\chi_{207} (1, \cdot )$
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 \( 1 - 9.32T + 32T^{2} \)
5 \( 1 + 99.7T + 3.12e3T^{2} \)
7 \( 1 - 125.T + 1.68e4T^{2} \)
11 \( 1 + 177.T + 1.61e5T^{2} \)
13 \( 1 + 919.T + 3.71e5T^{2} \)
17 \( 1 + 1.56e3T + 1.41e6T^{2} \)
19 \( 1 - 447.T + 2.47e6T^{2} \)
29 \( 1 - 1.29e3T + 2.05e7T^{2} \)
31 \( 1 + 6.65e3T + 2.86e7T^{2} \)
37 \( 1 + 8.05e3T + 6.93e7T^{2} \)
41 \( 1 + 1.79e4T + 1.15e8T^{2} \)
43 \( 1 - 1.07e4T + 1.47e8T^{2} \)
47 \( 1 - 1.78e4T + 2.29e8T^{2} \)
53 \( 1 - 2.03e4T + 4.18e8T^{2} \)
59 \( 1 + 2.84e4T + 7.14e8T^{2} \)
61 \( 1 - 1.62e4T + 8.44e8T^{2} \)
67 \( 1 - 5.41e4T + 1.35e9T^{2} \)
71 \( 1 + 4.25e4T + 1.80e9T^{2} \)
73 \( 1 - 4.09e4T + 2.07e9T^{2} \)
79 \( 1 + 2.47e4T + 3.07e9T^{2} \)
83 \( 1 - 3.08e4T + 3.93e9T^{2} \)
89 \( 1 - 6.65e4T + 5.58e9T^{2} \)
97 \( 1 + 1.04e5T + 8.58e9T^{2} \)
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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

−11.45753341952795246274899398902, −10.70920981130083802671423911923, −8.741713999613928393615283364823, −7.56993171792478183357468199172, −6.96736703447452818538092760264, −5.16103936442287838475006853797, −4.60936165108928591535315987970, −3.63092596899130499872450297993, −2.33820534735904312861278639632, 0, 2.33820534735904312861278639632, 3.63092596899130499872450297993, 4.60936165108928591535315987970, 5.16103936442287838475006853797, 6.96736703447452818538092760264, 7.56993171792478183357468199172, 8.741713999613928393615283364823, 10.70920981130083802671423911923, 11.45753341952795246274899398902

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