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Properties

Label 2-222180-1.1-c1-0-10
Degree $2$
Conductor $222180$
Sign $1$
Analytic cond. $1774.11$
Root an. cond. $42.1202$
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 − 5-s − 7-s + 9-s + 4·11-s + 13-s − 15-s + 3·17-s + 6·19-s − 21-s + 25-s + 27-s + 29-s + 8·31-s + 4·33-s + 35-s + 4·37-s + 39-s − 10·41-s − 2·43-s − 45-s + 7·47-s + 49-s + 3·51-s + 12·53-s − 4·55-s + 6·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.727·17-s + 1.37·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.185·29-s + 1.43·31-s + 0.696·33-s + 0.169·35-s + 0.657·37-s + 0.160·39-s − 1.56·41-s − 0.304·43-s − 0.149·45-s + 1.02·47-s + 1/7·49-s + 0.420·51-s + 1.64·53-s − 0.539·55-s + 0.794·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(222180\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1774.11\)
Root analytic conductor: \(42.1202\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 222180,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.840415170\)
\(L(\frac12)\) \(\approx\) \(4.840415170\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.16409879079147, −12.33190992738732, −12.04141266923792, −11.54877842182126, −11.42148810833264, −10.42842614745462, −10.01611459527635, −9.817016583778209, −9.127431651278894, −8.722390906205930, −8.308855090093854, −7.798962357598750, −7.197263240223440, −6.855973982585916, −6.399056469757688, −5.662257177366678, −5.288124934403964, −4.528969041555179, −3.972784328214519, −3.633199319901653, −3.066732661808739, −2.595764764193625, −1.763605777298341, −1.017349574252424, −0.7238992527731811, 0.7238992527731811, 1.017349574252424, 1.763605777298341, 2.595764764193625, 3.066732661808739, 3.633199319901653, 3.972784328214519, 4.528969041555179, 5.288124934403964, 5.662257177366678, 6.399056469757688, 6.855973982585916, 7.197263240223440, 7.798962357598750, 8.308855090093854, 8.722390906205930, 9.127431651278894, 9.817016583778209, 10.01611459527635, 10.42842614745462, 11.42148810833264, 11.54877842182126, 12.04141266923792, 12.33190992738732, 13.16409879079147

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