Properties

Label 2-1170-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s + 10-s − 13-s + 16-s + 6·17-s + 20-s + 4·23-s + 25-s − 26-s + 10·29-s + 32-s + 6·34-s − 6·37-s + 40-s − 2·41-s − 4·43-s + 4·46-s − 7·49-s + 50-s − 52-s + 6·53-s + 10·58-s + 6·61-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 1.85·29-s + 0.176·32-s + 1.02·34-s − 0.986·37-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + 0.589·46-s − 49-s + 0.141·50-s − 0.138·52-s + 0.824·53-s + 1.31·58-s + 0.768·61-s + 1/8·64-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: $\chi_{1170} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.880913927\)
\(L(\frac12)\) \(\approx\) \(2.880913927\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−10.07123462831113686449175462085, −8.955383165969649286288406642539, −8.062771471643955273434189645297, −7.11791279895358735884585299226, −6.36199631841087100215380294304, −5.39759482678390501023193939738, −4.78086243570270981436649484362, −3.52270970074791721181391566206, −2.67714551530238471219197858825, −1.30131976065987787395283637597, 1.30131976065987787395283637597, 2.67714551530238471219197858825, 3.52270970074791721181391566206, 4.78086243570270981436649484362, 5.39759482678390501023193939738, 6.36199631841087100215380294304, 7.11791279895358735884585299226, 8.062771471643955273434189645297, 8.955383165969649286288406642539, 10.07123462831113686449175462085

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