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Properties

Label 2-222180-1.1-c1-0-16
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 − 5·11-s + 4·13-s − 15-s + 6·17-s − 19-s + 21-s + 25-s + 27-s − 10·29-s + 11·31-s − 5·33-s − 35-s − 2·37-s + 4·39-s + 7·41-s + 4·43-s − 45-s − 10·47-s + 49-s + 6·51-s − 6·53-s + 5·55-s − 57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s − 0.229·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 1.97·31-s − 0.870·33-s − 0.169·35-s − 0.328·37-s + 0.640·39-s + 1.09·41-s + 0.609·43-s − 0.149·45-s − 1.45·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.674·55-s − 0.132·57-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: \(1\)
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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−13.16257688965670, −12.89284782381288, −12.26735097132400, −11.85334236017099, −11.26701812769991, −10.79768859979435, −10.49852193827819, −9.923107984176689, −9.452670062491396, −8.902651344407097, −8.321087837631249, −7.968874615444256, −7.664978378920830, −7.305525693863385, −6.420201715604676, −5.956765549191955, −5.511143881100389, −4.841011039806324, −4.451129050378699, −3.772197011147142, −3.180629398924565, −2.939430301456352, −2.129571130024425, −1.505839160387048, −0.8753127803494905, 0, 0.8753127803494905, 1.505839160387048, 2.129571130024425, 2.939430301456352, 3.180629398924565, 3.772197011147142, 4.451129050378699, 4.841011039806324, 5.511143881100389, 5.956765549191955, 6.420201715604676, 7.305525693863385, 7.664978378920830, 7.968874615444256, 8.321087837631249, 8.902651344407097, 9.452670062491396, 9.923107984176689, 10.49852193827819, 10.79768859979435, 11.26701812769991, 11.85334236017099, 12.26735097132400, 12.89284782381288, 13.16257688965670

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