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Properties

Label 2-32340-1.1-c1-0-0
Degree $2$
Conductor $32340$
Sign $1$
Analytic cond. $258.236$
Root an. cond. $16.0697$
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 + 9-s − 11-s + 13-s − 15-s − 2·19-s − 9·23-s + 25-s − 27-s − 9·29-s − 8·31-s + 33-s + 2·37-s − 39-s − 3·41-s − 43-s + 45-s − 9·47-s + 9·53-s − 55-s + 2·57-s − 12·59-s + 10·61-s + 65-s − 4·67-s + 9·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.258·15-s − 0.458·19-s − 1.87·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s − 1.43·31-s + 0.174·33-s + 0.328·37-s − 0.160·39-s − 0.468·41-s − 0.152·43-s + 0.149·45-s − 1.31·47-s + 1.23·53-s − 0.134·55-s + 0.264·57-s − 1.56·59-s + 1.28·61-s + 0.124·65-s − 0.488·67-s + 1.08·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32340\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(258.236\)
Root analytic conductor: \(16.0697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8429207953\)
\(L(\frac12)\) \(\approx\) \(0.8429207953\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
good13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.87232816559781, −14.67662276644426, −13.95376309424057, −13.24861399454057, −13.10027945469197, −12.36717176445280, −11.89914865328793, −11.20500181403001, −10.90407126207223, −10.17489886962854, −9.807153330215664, −9.210172435105681, −8.536721871579083, −7.939916281755581, −7.350391249446898, −6.735519474099102, −6.056107017362203, −5.642951050692465, −5.164597103235858, −4.227994266453043, −3.839130340695026, −2.968361859232287, −1.941810202280351, −1.677069736226070, −0.3434400209137580, 0.3434400209137580, 1.677069736226070, 1.941810202280351, 2.968361859232287, 3.839130340695026, 4.227994266453043, 5.164597103235858, 5.642951050692465, 6.056107017362203, 6.735519474099102, 7.350391249446898, 7.939916281755581, 8.536721871579083, 9.210172435105681, 9.807153330215664, 10.17489886962854, 10.90407126207223, 11.20500181403001, 11.89914865328793, 12.36717176445280, 13.10027945469197, 13.24861399454057, 13.95376309424057, 14.67662276644426, 14.87232816559781

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