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Properties

Label 2-105e2-1.1-c1-0-35
Degree $2$
Conductor $11025$
Sign $-1$
Analytic cond. $88.0350$
Root an. cond. $9.38270$
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  + 2-s − 4-s − 3·8-s − 6·13-s − 16-s − 2·17-s + 8·19-s + 8·23-s − 6·26-s + 2·29-s − 4·31-s + 5·32-s − 2·34-s + 2·37-s + 8·38-s − 6·41-s − 4·43-s + 8·46-s − 8·47-s + 6·52-s + 10·53-s + 2·58-s + 4·59-s + 2·61-s − 4·62-s + 7·64-s − 4·67-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1.66·13-s − 1/4·16-s − 0.485·17-s + 1.83·19-s + 1.66·23-s − 1.17·26-s + 0.371·29-s − 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.328·37-s + 1.29·38-s − 0.937·41-s − 0.609·43-s + 1.17·46-s − 1.16·47-s + 0.832·52-s + 1.37·53-s + 0.262·58-s + 0.520·59-s + 0.256·61-s − 0.508·62-s + 7/8·64-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11025\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(88.0350\)
Root analytic conductor: \(9.38270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11025,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−16.72237721915346, −16.26165633230791, −15.35211768463677, −14.92358325157380, −14.60859600193107, −13.78590832522480, −13.47483419320935, −12.82028051186910, −12.26600392062564, −11.76157312455521, −11.20975416035185, −10.29224097768517, −9.587978706342278, −9.372712987930725, −8.564417779484274, −7.844750921391283, −7.048775273905974, −6.691265488473300, −5.476489495379934, −5.179223014180048, −4.701886696483178, −3.763628836400713, −3.070859664274092, −2.453991589086128, −1.127880531938779, 0, 1.127880531938779, 2.453991589086128, 3.070859664274092, 3.763628836400713, 4.701886696483178, 5.179223014180048, 5.476489495379934, 6.691265488473300, 7.048775273905974, 7.844750921391283, 8.564417779484274, 9.372712987930725, 9.587978706342278, 10.29224097768517, 11.20975416035185, 11.76157312455521, 12.26600392062564, 12.82028051186910, 13.47483419320935, 13.78590832522480, 14.60859600193107, 14.92358325157380, 15.35211768463677, 16.26165633230791, 16.72237721915346

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