Properties

Label 2-207-1.1-c1-0-7
Degree $2$
Conductor $207$
Sign $1$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·2-s + 3.82·4-s + 0.585·5-s − 3.41·7-s + 4.41·8-s + 1.41·10-s − 2.82·11-s − 8.24·14-s + 2.99·16-s + 7.41·17-s − 6.24·19-s + 2.24·20-s − 6.82·22-s + 23-s − 4.65·25-s − 13.0·28-s + 8.48·29-s + 8.48·31-s − 1.58·32-s + 17.8·34-s − 2·35-s − 4.82·37-s − 15.0·38-s + 2.58·40-s − 1.65·41-s − 1.75·43-s − 10.8·44-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.91·4-s + 0.261·5-s − 1.29·7-s + 1.56·8-s + 0.447·10-s − 0.852·11-s − 2.20·14-s + 0.749·16-s + 1.79·17-s − 1.43·19-s + 0.501·20-s − 1.45·22-s + 0.208·23-s − 0.931·25-s − 2.47·28-s + 1.57·29-s + 1.52·31-s − 0.280·32-s + 3.06·34-s − 0.338·35-s − 0.793·37-s − 2.44·38-s + 0.408·40-s − 0.258·41-s − 0.267·43-s − 1.63·44-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 207,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.645501427\)
\(L(\frac12)\) \(\approx\) \(2.645501427\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 - 2.41T + 2T^{2} \)
5 \( 1 - 0.585T + 5T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.41T + 17T^{2} \)
19 \( 1 + 6.24T + 19T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 + 4.82T + 37T^{2} \)
41 \( 1 + 1.65T + 41T^{2} \)
43 \( 1 + 1.75T + 43T^{2} \)
47 \( 1 + 0.343T + 47T^{2} \)
53 \( 1 + 5.07T + 53T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + 0.828T + 61T^{2} \)
67 \( 1 - 8.58T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 - 6.82T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−12.53143303684974527068627997510, −12.00574886498953030043687201754, −10.55121913261116891635483991909, −9.841814834630124358561652039511, −8.140672538903766441421366811653, −6.72003367824527123537023177594, −6.01545013893860363354515947754, −4.96020071882048629169365567112, −3.60321318675161354972819179117, −2.62931122275476910363783739075, 2.62931122275476910363783739075, 3.60321318675161354972819179117, 4.96020071882048629169365567112, 6.01545013893860363354515947754, 6.72003367824527123537023177594, 8.140672538903766441421366811653, 9.841814834630124358561652039511, 10.55121913261116891635483991909, 12.00574886498953030043687201754, 12.53143303684974527068627997510

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