Properties

Label 2-207-1.1-c5-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.29·2-s − 30.3·4-s − 59.1·5-s − 213.·7-s − 80.4·8-s − 76.2·10-s − 126.·11-s − 884.·13-s − 275.·14-s + 866.·16-s + 1.17e3·17-s − 1.86e3·19-s + 1.79e3·20-s − 163.·22-s − 529·23-s + 369.·25-s − 1.14e3·26-s + 6.47e3·28-s + 6.78e3·29-s − 5.14e3·31-s + 3.69e3·32-s + 1.52e3·34-s + 1.26e4·35-s + 5.13e3·37-s − 2.40e3·38-s + 4.75e3·40-s − 1.24e4·41-s + ⋯
L(s)  = 1  + 0.228·2-s − 0.947·4-s − 1.05·5-s − 1.64·7-s − 0.444·8-s − 0.241·10-s − 0.315·11-s − 1.45·13-s − 0.375·14-s + 0.846·16-s + 0.990·17-s − 1.18·19-s + 1.00·20-s − 0.0719·22-s − 0.208·23-s + 0.118·25-s − 0.331·26-s + 1.55·28-s + 1.49·29-s − 0.961·31-s + 0.637·32-s + 0.225·34-s + 1.74·35-s + 0.616·37-s − 0.270·38-s + 0.469·40-s − 1.15·41-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: \(0\)
Selberg data: \((2,\ 207,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2628667994\)
\(L(\frac12)\) \(\approx\) \(0.2628667994\)
\(L(\frac{7}{2})\) 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 \( 1 - 1.29T + 32T^{2} \)
5 \( 1 + 59.1T + 3.12e3T^{2} \)
7 \( 1 + 213.T + 1.68e4T^{2} \)
11 \( 1 + 126.T + 1.61e5T^{2} \)
13 \( 1 + 884.T + 3.71e5T^{2} \)
17 \( 1 - 1.17e3T + 1.41e6T^{2} \)
19 \( 1 + 1.86e3T + 2.47e6T^{2} \)
29 \( 1 - 6.78e3T + 2.05e7T^{2} \)
31 \( 1 + 5.14e3T + 2.86e7T^{2} \)
37 \( 1 - 5.13e3T + 6.93e7T^{2} \)
41 \( 1 + 1.24e4T + 1.15e8T^{2} \)
43 \( 1 - 4.19e3T + 1.47e8T^{2} \)
47 \( 1 + 2.30e4T + 2.29e8T^{2} \)
53 \( 1 + 2.51e4T + 4.18e8T^{2} \)
59 \( 1 - 3.71e4T + 7.14e8T^{2} \)
61 \( 1 - 2.64e4T + 8.44e8T^{2} \)
67 \( 1 + 5.43e4T + 1.35e9T^{2} \)
71 \( 1 + 3.56e4T + 1.80e9T^{2} \)
73 \( 1 - 3.39e4T + 2.07e9T^{2} \)
79 \( 1 + 7.66e4T + 3.07e9T^{2} \)
83 \( 1 - 9.66e4T + 3.93e9T^{2} \)
89 \( 1 + 3.00e4T + 5.58e9T^{2} \)
97 \( 1 + 1.46e4T + 8.58e9T^{2} \)
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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

−11.88985588527675319707324387570, −10.22924458657577991475564247962, −9.696759384747092465244397555437, −8.520397999797927016256298347919, −7.52315990871588891008226601862, −6.33336576564512094649636509210, −4.98482712103859096299220930750, −3.88163664816850123492767152290, −2.94139719933452043445025123526, −0.28653815402937130228886915298, 0.28653815402937130228886915298, 2.94139719933452043445025123526, 3.88163664816850123492767152290, 4.98482712103859096299220930750, 6.33336576564512094649636509210, 7.52315990871588891008226601862, 8.520397999797927016256298347919, 9.696759384747092465244397555437, 10.22924458657577991475564247962, 11.88985588527675319707324387570

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