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Properties

Label 2-203280-1.1-c1-0-166
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 7-s + 9-s + 6·13-s − 15-s + 7·17-s − 5·19-s − 21-s + 23-s + 25-s − 27-s + 5·29-s + 8·31-s + 35-s − 2·37-s − 6·39-s − 12·41-s − 11·43-s + 45-s − 8·47-s + 49-s − 7·51-s − 11·53-s + 5·57-s + 5·59-s − 7·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.258·15-s + 1.69·17-s − 1.14·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s − 0.960·39-s − 1.87·41-s − 1.67·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.980·51-s − 1.51·53-s + 0.662·57-s + 0.650·59-s − 0.896·61-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: \(1\)
Selberg data: \((2,\ 203280,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 \)
good13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−13.15741616073477, −12.96638180419343, −12.22603603034083, −11.79790413172151, −11.54024147012784, −10.90829702625701, −10.32006585021388, −10.21235738369686, −9.727704315170763, −8.854633134469084, −8.545703853261436, −8.121195717594580, −7.679622985724862, −6.771465680987143, −6.461991605814967, −6.160457146388813, −5.512504686383714, −5.015812416250308, −4.597356440039198, −3.923119335489768, −3.217164962680315, −2.989168193105052, −1.808446748193470, −1.499106596258253, −0.9576039701474366, 0, 0.9576039701474366, 1.499106596258253, 1.808446748193470, 2.989168193105052, 3.217164962680315, 3.923119335489768, 4.597356440039198, 5.015812416250308, 5.512504686383714, 6.160457146388813, 6.461991605814967, 6.771465680987143, 7.679622985724862, 8.121195717594580, 8.545703853261436, 8.854633134469084, 9.727704315170763, 10.21235738369686, 10.32006585021388, 10.90829702625701, 11.54024147012784, 11.79790413172151, 12.22603603034083, 12.96638180419343, 13.15741616073477

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