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Properties

Label 2-222180-1.1-c1-0-17
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 7-s + 9-s + 2·11-s − 15-s + 2·17-s + 2·19-s − 21-s + 25-s − 27-s − 10·29-s − 2·31-s − 2·33-s + 35-s + 2·37-s + 2·41-s + 4·43-s + 45-s + 49-s − 2·51-s − 8·53-s + 2·55-s − 2·57-s + 12·59-s + 2·61-s + 63-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.258·15-s + 0.485·17-s + 0.458·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.359·31-s − 0.348·33-s + 0.169·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s − 1.09·53-s + 0.269·55-s − 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.125·63-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: \(1\)
Selberg data: \((2,\ 222180,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.10120173366924, −12.86400275279787, −12.10872343181905, −11.82236488529623, −11.31673173155301, −10.96360811632236, −10.46282915479781, −9.833384025810858, −9.576089196612539, −9.020333888713592, −8.592490809730210, −7.886206150682608, −7.338759400400060, −7.172025559657447, −6.332364768667741, −5.967272519559699, −5.516587446116533, −5.067831783887472, −4.461590856184134, −3.877732117425375, −3.436170091699960, −2.656995786515853, −1.985361471109561, −1.448422675948913, −0.8850186735249406, 0, 0.8850186735249406, 1.448422675948913, 1.985361471109561, 2.656995786515853, 3.436170091699960, 3.877732117425375, 4.461590856184134, 5.067831783887472, 5.516587446116533, 5.967272519559699, 6.332364768667741, 7.172025559657447, 7.338759400400060, 7.886206150682608, 8.592490809730210, 9.020333888713592, 9.576089196612539, 9.833384025810858, 10.46282915479781, 10.96360811632236, 11.31673173155301, 11.82236488529623, 12.10872343181905, 12.86400275279787, 13.10120173366924

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