Properties

Label 2-23064-1.1-c1-0-1
Degree $2$
Conductor $23064$
Sign $1$
Analytic cond. $184.166$
Root an. cond. $13.5708$
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  + 3-s − 2·5-s + 9-s − 4·11-s + 2·13-s − 2·15-s − 2·17-s − 4·19-s + 8·23-s − 25-s + 27-s − 6·29-s − 4·33-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s − 2·45-s − 7·49-s − 2·51-s + 2·53-s + 8·55-s − 4·57-s + 4·59-s + 2·61-s − 4·65-s − 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s − 49-s − 0.280·51-s + 0.274·53-s + 1.07·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.496·65-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23064\)    =    \(2^{3} \cdot 3 \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(184.166\)
Root analytic conductor: \(13.5708\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{23064} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 23064,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.211078806\)
\(L(\frac12)\) \(\approx\) \(1.211078806\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
31 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.37385324610571, −15.04804195947055, −14.61860326122116, −13.70376142350147, −13.18278191454258, −13.03597571529414, −12.26887691737149, −11.63481856629517, −10.94127515074543, −10.75515503869090, −9.987160648524284, −9.292937957880942, −8.610031784173350, −8.331007903071827, −7.701604238582486, −7.100712992765364, −6.628560631102336, −5.691700537378752, −5.008346902079851, −4.468428290810396, −3.604606608855534, −3.260223147879841, −2.366561878101926, −1.653728411581996, −0.4142120890422641, 0.4142120890422641, 1.653728411581996, 2.366561878101926, 3.260223147879841, 3.604606608855534, 4.468428290810396, 5.008346902079851, 5.691700537378752, 6.628560631102336, 7.100712992765364, 7.701604238582486, 8.331007903071827, 8.610031784173350, 9.292937957880942, 9.987160648524284, 10.75515503869090, 10.94127515074543, 11.63481856629517, 12.26887691737149, 13.03597571529414, 13.18278191454258, 13.70376142350147, 14.61860326122116, 15.04804195947055, 15.37385324610571

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