Properties

Label 2-1170-1.1-c1-0-13
Degree $2$
Conductor $1170$
Sign $-1$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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 − 5-s − 8-s + 10-s − 4·11-s + 13-s + 16-s + 6·17-s + 4·19-s − 20-s + 4·22-s − 8·23-s + 25-s − 26-s − 6·29-s − 8·31-s − 32-s − 6·34-s − 10·37-s − 4·38-s + 40-s + 6·41-s + 4·43-s − 4·44-s + 8·46-s − 7·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s − 0.196·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 1.64·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s − 0.603·44-s + 1.17·46-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.402437326353945088461831893248, −8.490403204098803911613209618504, −7.54788789198252757941968322284, −7.45586944398122020573570806182, −5.88475078175345459986396322172, −5.36101206913072414201885147070, −3.89256533853822824063870308537, −2.98523621593750721047361308357, −1.63917888763865576740316873481, 0, 1.63917888763865576740316873481, 2.98523621593750721047361308357, 3.89256533853822824063870308537, 5.36101206913072414201885147070, 5.88475078175345459986396322172, 7.45586944398122020573570806182, 7.54788789198252757941968322284, 8.490403204098803911613209618504, 9.402437326353945088461831893248

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