Properties

Label 2-201-1.1-c3-0-24
Degree $2$
Conductor $201$
Sign $1$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.42·2-s + 3·3-s + 21.4·4-s − 20.0·5-s + 16.2·6-s + 33.9·7-s + 73.0·8-s + 9·9-s − 108.·10-s − 1.17·11-s + 64.3·12-s − 21.8·13-s + 184.·14-s − 60.1·15-s + 225.·16-s − 115.·17-s + 48.8·18-s + 8.20·19-s − 430.·20-s + 101.·21-s − 6.36·22-s + 44.0·23-s + 219.·24-s + 276.·25-s − 118.·26-s + 27·27-s + 728.·28-s + ⋯
L(s)  = 1  + 1.91·2-s + 0.577·3-s + 2.68·4-s − 1.79·5-s + 1.10·6-s + 1.83·7-s + 3.23·8-s + 0.333·9-s − 3.43·10-s − 0.0321·11-s + 1.54·12-s − 0.466·13-s + 3.51·14-s − 1.03·15-s + 3.51·16-s − 1.65·17-s + 0.639·18-s + 0.0990·19-s − 4.80·20-s + 1.05·21-s − 0.0616·22-s + 0.399·23-s + 1.86·24-s + 2.21·25-s − 0.894·26-s + 0.192·27-s + 4.91·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $1$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.793017694\)
\(L(\frac12)\) \(\approx\) \(5.793017694\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
67 \( 1 - 67T \)
good2 \( 1 - 5.42T + 8T^{2} \)
5 \( 1 + 20.0T + 125T^{2} \)
7 \( 1 - 33.9T + 343T^{2} \)
11 \( 1 + 1.17T + 1.33e3T^{2} \)
13 \( 1 + 21.8T + 2.19e3T^{2} \)
17 \( 1 + 115.T + 4.91e3T^{2} \)
19 \( 1 - 8.20T + 6.85e3T^{2} \)
23 \( 1 - 44.0T + 1.21e4T^{2} \)
29 \( 1 + 169.T + 2.43e4T^{2} \)
31 \( 1 - 14.2T + 2.97e4T^{2} \)
37 \( 1 + 112.T + 5.06e4T^{2} \)
41 \( 1 - 7.90T + 6.89e4T^{2} \)
43 \( 1 + 448.T + 7.95e4T^{2} \)
47 \( 1 + 342.T + 1.03e5T^{2} \)
53 \( 1 + 39.9T + 1.48e5T^{2} \)
59 \( 1 + 126.T + 2.05e5T^{2} \)
61 \( 1 + 308.T + 2.26e5T^{2} \)
71 \( 1 - 1.09e3T + 3.57e5T^{2} \)
73 \( 1 - 225.T + 3.89e5T^{2} \)
79 \( 1 + 31.5T + 4.93e5T^{2} \)
83 \( 1 + 514.T + 5.71e5T^{2} \)
89 \( 1 - 873.T + 7.04e5T^{2} \)
97 \( 1 - 309.T + 9.12e5T^{2} \)
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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

−12.02090755567024226875328852472, −11.34208455693682143712183931252, −10.90867909528766619431832099583, −8.466822323971601035010196934307, −7.66584694259418470426444306978, −6.90380395612456598058860020028, −4.95942087290542895856727499995, −4.48273775840276456244405313332, −3.47013417978219319118816036044, −1.99518250845458040922406622160, 1.99518250845458040922406622160, 3.47013417978219319118816036044, 4.48273775840276456244405313332, 4.95942087290542895856727499995, 6.90380395612456598058860020028, 7.66584694259418470426444306978, 8.466822323971601035010196934307, 10.90867909528766619431832099583, 11.34208455693682143712183931252, 12.02090755567024226875328852472

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