Properties

Label 2-207-1.1-c7-0-14
Degree $2$
Conductor $207$
Sign $1$
Analytic cond. $64.6637$
Root an. cond. $8.04137$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.570·2-s − 127.·4-s − 128.·5-s + 1.73e3·7-s − 145.·8-s − 73.5·10-s − 5.84e3·11-s − 2.25e3·13-s + 989.·14-s + 1.62e4·16-s − 7.81e3·17-s + 2.46e4·19-s + 1.64e4·20-s − 3.33e3·22-s + 1.21e4·23-s − 6.15e4·25-s − 1.28e3·26-s − 2.21e5·28-s − 1.22e5·29-s − 1.05e5·31-s + 2.79e4·32-s − 4.46e3·34-s − 2.23e5·35-s + 3.89e5·37-s + 1.40e4·38-s + 1.88e4·40-s − 8.58e5·41-s + ⋯
L(s)  = 1  + 0.0504·2-s − 0.997·4-s − 0.461·5-s + 1.90·7-s − 0.100·8-s − 0.0232·10-s − 1.32·11-s − 0.284·13-s + 0.0963·14-s + 0.992·16-s − 0.385·17-s + 0.825·19-s + 0.460·20-s − 0.0668·22-s + 0.208·23-s − 0.787·25-s − 0.0143·26-s − 1.90·28-s − 0.933·29-s − 0.636·31-s + 0.150·32-s − 0.0194·34-s − 0.880·35-s + 1.26·37-s + 0.0416·38-s + 0.0464·40-s − 1.94·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(64.6637\)
Root analytic conductor: \(8.04137\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.492209967\)
\(L(\frac12)\) \(\approx\) \(1.492209967\)
\(L(\frac{9}{2})\) 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 \( 1 - 1.21e4T \)
good2 \( 1 - 0.570T + 128T^{2} \)
5 \( 1 + 128.T + 7.81e4T^{2} \)
7 \( 1 - 1.73e3T + 8.23e5T^{2} \)
11 \( 1 + 5.84e3T + 1.94e7T^{2} \)
13 \( 1 + 2.25e3T + 6.27e7T^{2} \)
17 \( 1 + 7.81e3T + 4.10e8T^{2} \)
19 \( 1 - 2.46e4T + 8.93e8T^{2} \)
29 \( 1 + 1.22e5T + 1.72e10T^{2} \)
31 \( 1 + 1.05e5T + 2.75e10T^{2} \)
37 \( 1 - 3.89e5T + 9.49e10T^{2} \)
41 \( 1 + 8.58e5T + 1.94e11T^{2} \)
43 \( 1 + 5.97e4T + 2.71e11T^{2} \)
47 \( 1 - 7.25e5T + 5.06e11T^{2} \)
53 \( 1 - 9.13e5T + 1.17e12T^{2} \)
59 \( 1 - 1.83e6T + 2.48e12T^{2} \)
61 \( 1 + 4.45e4T + 3.14e12T^{2} \)
67 \( 1 - 5.60e5T + 6.06e12T^{2} \)
71 \( 1 - 1.76e6T + 9.09e12T^{2} \)
73 \( 1 - 2.12e6T + 1.10e13T^{2} \)
79 \( 1 - 6.56e6T + 1.92e13T^{2} \)
83 \( 1 - 5.02e6T + 2.71e13T^{2} \)
89 \( 1 - 1.13e7T + 4.42e13T^{2} \)
97 \( 1 + 1.97e6T + 8.07e13T^{2} \)
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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

−11.17986541017200755780619801655, −10.20669089394825336954556983772, −8.974597853825621815144878038287, −7.997858973742598590408696134596, −7.57201844657041479695896196308, −5.40585604151620499372543288006, −4.92677527727442299370009753862, −3.80299041342963377804675897181, −2.10341829257695837041077154735, −0.64977361599542918974539890362, 0.64977361599542918974539890362, 2.10341829257695837041077154735, 3.80299041342963377804675897181, 4.92677527727442299370009753862, 5.40585604151620499372543288006, 7.57201844657041479695896196308, 7.997858973742598590408696134596, 8.974597853825621815144878038287, 10.20669089394825336954556983772, 11.17986541017200755780619801655

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