Properties

Degree $2$
Conductor $221067$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s + 2·13-s + 14-s − 16-s + 2·17-s + 4·19-s − 2·20-s − 25-s − 2·26-s + 28-s + 29-s − 8·31-s − 5·32-s − 2·34-s − 2·35-s − 10·37-s − 4·38-s + 6·40-s − 6·41-s − 12·43-s + 8·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.447·20-s − 1/5·25-s − 0.392·26-s + 0.188·28-s + 0.185·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s − 0.338·35-s − 1.64·37-s − 0.648·38-s + 0.948·40-s − 0.937·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221067\)    =    \(3^{2} \cdot 7 \cdot 11^{2} \cdot 29\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{221067} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 221067,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9389620303\)
\(L(\frac12)\) \(\approx\) \(0.9389620303\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
29 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.06393181021712, −12.56956131361046, −12.01938416884210, −11.58438560186048, −10.81695719543798, −10.51752023996008, −10.10763000430194, −9.537489968398973, −9.371287210053131, −8.816584640943809, −8.366069453481992, −7.854846138152870, −7.307720193273748, −6.842198827474569, −6.289737605355255, −5.636948576421212, −5.274519895828632, −4.893228850780636, −4.045454541889537, −3.471118402192117, −3.187695161619867, −2.150232119159776, −1.649401491917834, −1.217577690466507, −0.3188949417562131, 0.3188949417562131, 1.217577690466507, 1.649401491917834, 2.150232119159776, 3.187695161619867, 3.471118402192117, 4.045454541889537, 4.893228850780636, 5.274519895828632, 5.636948576421212, 6.289737605355255, 6.842198827474569, 7.307720193273748, 7.854846138152870, 8.366069453481992, 8.816584640943809, 9.371287210053131, 9.537489968398973, 10.10763000430194, 10.51752023996008, 10.81695719543798, 11.58438560186048, 12.01938416884210, 12.56956131361046, 13.06393181021712

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