Properties

Label 2-135-1.1-c3-0-10
Degree $2$
Conductor $135$
Sign $-1$
Analytic cond. $7.96525$
Root an. cond. $2.82227$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·4-s + 5·5-s + 24·8-s − 10·10-s + 10·11-s − 80·13-s − 16·16-s + 7·17-s − 113·19-s − 20·20-s − 20·22-s − 81·23-s + 25·25-s + 160·26-s − 220·29-s − 189·31-s − 160·32-s − 14·34-s + 170·37-s + 226·38-s + 120·40-s − 130·41-s + 10·43-s − 40·44-s + 162·46-s + 160·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 0.274·11-s − 1.70·13-s − 1/4·16-s + 0.0998·17-s − 1.36·19-s − 0.223·20-s − 0.193·22-s − 0.734·23-s + 1/5·25-s + 1.20·26-s − 1.40·29-s − 1.09·31-s − 0.883·32-s − 0.0706·34-s + 0.755·37-s + 0.964·38-s + 0.474·40-s − 0.495·41-s + 0.0354·43-s − 0.137·44-s + 0.519·46-s + 0.496·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $-1$
Analytic conductor: \(7.96525\)
Root analytic conductor: \(2.82227\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{135} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 135,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
good2 \( 1 + p T + p^{3} T^{2} \)
7 \( 1 + p^{3} T^{2} \)
11 \( 1 - 10 T + p^{3} T^{2} \)
13 \( 1 + 80 T + p^{3} T^{2} \)
17 \( 1 - 7 T + p^{3} T^{2} \)
19 \( 1 + 113 T + p^{3} T^{2} \)
23 \( 1 + 81 T + p^{3} T^{2} \)
29 \( 1 + 220 T + p^{3} T^{2} \)
31 \( 1 + 189 T + p^{3} T^{2} \)
37 \( 1 - 170 T + p^{3} T^{2} \)
41 \( 1 + 130 T + p^{3} T^{2} \)
43 \( 1 - 10 T + p^{3} T^{2} \)
47 \( 1 - 160 T + p^{3} T^{2} \)
53 \( 1 - 631 T + p^{3} T^{2} \)
59 \( 1 + 560 T + p^{3} T^{2} \)
61 \( 1 - 229 T + p^{3} T^{2} \)
67 \( 1 - 750 T + p^{3} T^{2} \)
71 \( 1 - 890 T + p^{3} T^{2} \)
73 \( 1 + 890 T + p^{3} T^{2} \)
79 \( 1 + 27 T + p^{3} T^{2} \)
83 \( 1 - 429 T + p^{3} T^{2} \)
89 \( 1 + 750 T + p^{3} T^{2} \)
97 \( 1 + 1480 T + p^{3} T^{2} \)
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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

−12.36551583347829401166892986013, −10.93892655889584767736011162374, −9.906399364360154831934389757019, −9.272955834890351585263412951435, −8.108507795497751933118367780630, −7.03266955627461640977804320995, −5.44862821419945983963792486180, −4.18628886978418125446402177478, −2.04004367435515896269431503176, 0, 2.04004367435515896269431503176, 4.18628886978418125446402177478, 5.44862821419945983963792486180, 7.03266955627461640977804320995, 8.108507795497751933118367780630, 9.272955834890351585263412951435, 9.906399364360154831934389757019, 10.93892655889584767736011162374, 12.36551583347829401166892986013

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