Properties

Label 2-2233-1.1-c1-0-100
Degree $2$
Conductor $2233$
Sign $-1$
Analytic cond. $17.8305$
Root an. cond. $4.22262$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 7-s − 2·9-s − 11-s − 2·12-s + 2·13-s + 4·16-s + 6·17-s − 7·19-s + 21-s − 3·23-s − 5·25-s − 5·27-s − 2·28-s + 29-s − 4·31-s − 33-s + 4·36-s + 2·37-s + 2·39-s + 9·41-s − 4·43-s + 2·44-s − 9·47-s + 4·48-s + 49-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.577·12-s + 0.554·13-s + 16-s + 1.45·17-s − 1.60·19-s + 0.218·21-s − 0.625·23-s − 25-s − 0.962·27-s − 0.377·28-s + 0.185·29-s − 0.718·31-s − 0.174·33-s + 2/3·36-s + 0.328·37-s + 0.320·39-s + 1.40·41-s − 0.609·43-s + 0.301·44-s − 1.31·47-s + 0.577·48-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2233 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2233 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2233\)    =    \(7 \cdot 11 \cdot 29\)
Sign: $-1$
Analytic conductor: \(17.8305\)
Root analytic conductor: \(4.22262\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2233} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2233,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
11 \( 1 + T \)
29 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−8.449426738704921033911123355724, −8.211465775907911429013068789934, −7.42532714704336028183876296579, −5.95843087654965787151006295911, −5.62452236100427465906973804385, −4.44220881558004033390294560419, −3.78501230616501816766088607371, −2.84410792856064290796363321251, −1.59581542715941783964660566261, 0, 1.59581542715941783964660566261, 2.84410792856064290796363321251, 3.78501230616501816766088607371, 4.44220881558004033390294560419, 5.62452236100427465906973804385, 5.95843087654965787151006295911, 7.42532714704336028183876296579, 8.211465775907911429013068789934, 8.449426738704921033911123355724

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