Properties

Label 2-207-1.1-c7-0-58
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 17.8·2-s + 189.·4-s + 31.0·5-s − 1.24e3·7-s + 1.10e3·8-s + 553.·10-s + 939.·11-s + 1.01e4·13-s − 2.22e4·14-s − 4.66e3·16-s − 3.39e4·17-s + 1.87e4·19-s + 5.89e3·20-s + 1.67e4·22-s − 1.21e4·23-s − 7.71e4·25-s + 1.80e5·26-s − 2.36e5·28-s − 9.17e4·29-s − 1.87e5·31-s − 2.24e5·32-s − 6.05e5·34-s − 3.87e4·35-s − 1.12e5·37-s + 3.33e5·38-s + 3.41e4·40-s − 2.18e5·41-s + ⋯
L(s)  = 1  + 1.57·2-s + 1.48·4-s + 0.111·5-s − 1.37·7-s + 0.760·8-s + 0.174·10-s + 0.212·11-s + 1.28·13-s − 2.16·14-s − 0.284·16-s − 1.67·17-s + 0.626·19-s + 0.164·20-s + 0.335·22-s − 0.208·23-s − 0.987·25-s + 2.01·26-s − 2.03·28-s − 0.698·29-s − 1.12·31-s − 1.20·32-s − 2.64·34-s − 0.152·35-s − 0.364·37-s + 0.986·38-s + 0.0844·40-s − 0.495·41-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: $\chi_{207} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 207,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 - 17.8T + 128T^{2} \)
5 \( 1 - 31.0T + 7.81e4T^{2} \)
7 \( 1 + 1.24e3T + 8.23e5T^{2} \)
11 \( 1 - 939.T + 1.94e7T^{2} \)
13 \( 1 - 1.01e4T + 6.27e7T^{2} \)
17 \( 1 + 3.39e4T + 4.10e8T^{2} \)
19 \( 1 - 1.87e4T + 8.93e8T^{2} \)
29 \( 1 + 9.17e4T + 1.72e10T^{2} \)
31 \( 1 + 1.87e5T + 2.75e10T^{2} \)
37 \( 1 + 1.12e5T + 9.49e10T^{2} \)
41 \( 1 + 2.18e5T + 1.94e11T^{2} \)
43 \( 1 + 9.10e5T + 2.71e11T^{2} \)
47 \( 1 + 5.40e4T + 5.06e11T^{2} \)
53 \( 1 - 1.35e5T + 1.17e12T^{2} \)
59 \( 1 - 2.26e6T + 2.48e12T^{2} \)
61 \( 1 - 1.24e5T + 3.14e12T^{2} \)
67 \( 1 - 3.30e6T + 6.06e12T^{2} \)
71 \( 1 - 7.40e5T + 9.09e12T^{2} \)
73 \( 1 - 2.68e6T + 1.10e13T^{2} \)
79 \( 1 + 6.55e6T + 1.92e13T^{2} \)
83 \( 1 + 3.21e6T + 2.71e13T^{2} \)
89 \( 1 - 1.14e7T + 4.42e13T^{2} \)
97 \( 1 + 1.41e7T + 8.07e13T^{2} \)
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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

−11.04934545847610131342396837010, −9.723047774747086958302324534739, −8.714991068199041651226320537540, −6.93887458600869271107565468942, −6.30965392366812325065295618293, −5.39375865557537739891844954289, −3.96217389850277021219682826754, −3.37359029810187515472501938684, −2.01519326045091043598140080430, 0, 2.01519326045091043598140080430, 3.37359029810187515472501938684, 3.96217389850277021219682826754, 5.39375865557537739891844954289, 6.30965392366812325065295618293, 6.93887458600869271107565468942, 8.714991068199041651226320537540, 9.723047774747086958302324534739, 11.04934545847610131342396837010

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