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Properties

Label 2-203280-1.1-c1-0-31
Degree $2$
Conductor $203280$
Sign $1$
Analytic cond. $1623.19$
Root an. cond. $40.2889$
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 − 2·13-s + 15-s − 6·17-s − 4·19-s + 21-s − 4·23-s + 25-s − 27-s + 8·29-s + 2·31-s + 35-s − 8·37-s + 2·39-s + 10·41-s − 45-s + 8·47-s + 49-s + 6·51-s + 2·53-s + 4·57-s + 6·59-s + 6·61-s − 63-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.359·31-s + 0.169·35-s − 1.31·37-s + 0.320·39-s + 1.56·41-s − 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.274·53-s + 0.529·57-s + 0.781·59-s + 0.768·61-s − 0.125·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(203280\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1623.19\)
Root analytic conductor: \(40.2889\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 203280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.162699722\)
\(L(\frac12)\) \(\approx\) \(1.162699722\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.94683737835952, −12.38157438683269, −12.27435469096082, −11.65213146348382, −11.19005204682018, −10.67166729451896, −10.37058228082550, −9.845987881061258, −9.282802207054930, −8.700742537519449, −8.407758772644666, −7.762590319292948, −7.197469632143311, −6.717838140949434, −6.386648375206712, −5.879629400008956, −5.149285189052248, −4.741159961946694, −4.085940798353985, −3.922839736074170, −2.987837716198353, −2.322868232456752, −2.022681455801407, −0.8809178176888120, −0.3938279199318298, 0.3938279199318298, 0.8809178176888120, 2.022681455801407, 2.322868232456752, 2.987837716198353, 3.922839736074170, 4.085940798353985, 4.741159961946694, 5.149285189052248, 5.879629400008956, 6.386648375206712, 6.717838140949434, 7.197469632143311, 7.762590319292948, 8.407758772644666, 8.700742537519449, 9.282802207054930, 9.845987881061258, 10.37058228082550, 10.67166729451896, 11.19005204682018, 11.65213146348382, 12.27435469096082, 12.38157438683269, 12.94683737835952

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