Properties

Label 2-1110-1.1-c1-0-19
Degree $2$
Conductor $1110$
Sign $-1$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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 − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s + 9-s − 10-s − 5·11-s − 12-s − 2·13-s − 3·14-s − 15-s + 16-s − 7·17-s − 18-s − 2·19-s + 20-s − 3·21-s + 5·22-s + 4·23-s + 24-s + 25-s + 2·26-s − 27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s − 0.554·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.654·21-s + 1.06·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
37 \( 1 - T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−9.363922753414612521981306383391, −8.706783633055051794435143755747, −7.69264908599158201619263252174, −7.17751491453002082324977196302, −6.01505558146643513900988950471, −5.18928917671767552208400171306, −4.44657436788714417117593472562, −2.61172705280753441978196755354, −1.75806217276926209947046315627, 0, 1.75806217276926209947046315627, 2.61172705280753441978196755354, 4.44657436788714417117593472562, 5.18928917671767552208400171306, 6.01505558146643513900988950471, 7.17751491453002082324977196302, 7.69264908599158201619263252174, 8.706783633055051794435143755747, 9.363922753414612521981306383391

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