Properties

Label 2-110466-1.1-c1-0-24
Degree $2$
Conductor $110466$
Sign $1$
Analytic cond. $882.075$
Root an. cond. $29.6997$
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 + 4-s + 2·5-s + 8-s + 2·10-s + 4·11-s + 2·13-s + 16-s − 17-s + 2·20-s + 4·22-s − 25-s + 2·26-s − 10·29-s − 8·31-s + 32-s − 34-s + 2·37-s + 2·40-s + 10·41-s + 12·43-s + 4·44-s − 7·49-s − 50-s + 2·52-s + 6·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.392·26-s − 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.171·34-s + 0.328·37-s + 0.316·40-s + 1.56·41-s + 1.82·43-s + 0.603·44-s − 49-s − 0.141·50-s + 0.277·52-s + 0.824·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110466\)    =    \(2 \cdot 3^{2} \cdot 17 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(882.075\)
Root analytic conductor: \(29.6997\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 110466,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.271620138\)
\(L(\frac12)\) \(\approx\) \(6.271620138\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 + T \)
19 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
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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

−13.68802067007920, −13.13039568537319, −12.74256571542795, −12.46079983263071, −11.55194687767298, −11.30681232417555, −10.92568232414291, −10.30045165354897, −9.572392305131629, −9.279338043219121, −8.956923662646919, −8.141124922054058, −7.462754179058109, −7.136288068494579, −6.392184343987051, −5.981580777989327, −5.683280699831704, −5.064896537202601, −4.302377606787809, −3.752361546059185, −3.536038862029324, −2.422413043633208, −2.088224351291452, −1.438059030503147, −0.6843885735996437, 0.6843885735996437, 1.438059030503147, 2.088224351291452, 2.422413043633208, 3.536038862029324, 3.752361546059185, 4.302377606787809, 5.064896537202601, 5.683280699831704, 5.981580777989327, 6.392184343987051, 7.136288068494579, 7.462754179058109, 8.141124922054058, 8.956923662646919, 9.279338043219121, 9.572392305131629, 10.30045165354897, 10.92568232414291, 11.30681232417555, 11.55194687767298, 12.46079983263071, 12.74256571542795, 13.13039568537319, 13.68802067007920

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