Properties

Label 2-222-1.1-c1-0-4
Degree $2$
Conductor $222$
Sign $1$
Analytic cond. $1.77267$
Root an. cond. $1.33141$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 3·11-s + 12-s − 13-s − 14-s + 16-s − 3·17-s + 18-s − 7·19-s − 21-s + 3·22-s + 3·23-s + 24-s − 5·25-s − 26-s + 27-s − 28-s + 2·31-s + 32-s + 3·33-s − 3·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.60·19-s − 0.218·21-s + 0.639·22-s + 0.625·23-s + 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + 0.359·31-s + 0.176·32-s + 0.522·33-s − 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(222\)    =    \(2 \cdot 3 \cdot 37\)
Sign: $1$
Analytic conductor: \(1.77267\)
Root analytic conductor: \(1.33141\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 222,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.052995674\)
\(L(\frac12)\) \(\approx\) \(2.052995674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
37 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.45324066232302634795433200059, −11.47802135717016503590383469780, −10.39635634698152050574432699752, −9.300493923259646251478145783188, −8.333482702622619011476059157484, −6.99713618336569610002030002776, −6.20333969684185468580487444988, −4.62401395776704234514703639894, −3.61619630369357171767742863616, −2.15611933749619298935011997573, 2.15611933749619298935011997573, 3.61619630369357171767742863616, 4.62401395776704234514703639894, 6.20333969684185468580487444988, 6.99713618336569610002030002776, 8.333482702622619011476059157484, 9.300493923259646251478145783188, 10.39635634698152050574432699752, 11.47802135717016503590383469780, 12.45324066232302634795433200059

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