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Properties

Label 2-222180-1.1-c1-0-1
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

\(L(1)\) \(\approx\) \(2.275551691\)
\(L(\frac12)\) \(\approx\) \(2.275551691\)
\(L(\frac{3}{2})\) 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

−13.05353675162257, −12.57430402070952, −12.16054701280086, −11.56486042738703, −10.78770195322661, −10.72746462559683, −10.25172288303707, −9.658750095447669, −9.205181348090639, −8.554108093894044, −8.310883361279201, −7.961345285397208, −7.270130368158035, −6.696845267918584, −6.361069517150083, −5.730360783502987, −5.120898447600435, −4.642949728551803, −4.276749180649656, −3.476617172098531, −2.957706112208617, −2.296589970455643, −2.038407895687671, −1.289240449637952, −0.3876890132245247, 0.3876890132245247, 1.289240449637952, 2.038407895687671, 2.296589970455643, 2.957706112208617, 3.476617172098531, 4.276749180649656, 4.642949728551803, 5.120898447600435, 5.730360783502987, 6.361069517150083, 6.696845267918584, 7.270130368158035, 7.961345285397208, 8.310883361279201, 8.554108093894044, 9.205181348090639, 9.658750095447669, 10.25172288303707, 10.72746462559683, 10.78770195322661, 11.56486042738703, 12.16054701280086, 12.57430402070952, 13.05353675162257

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