Properties

Label 2-2200-1.1-c1-0-14
Degree $2$
Conductor $2200$
Sign $1$
Analytic cond. $17.5670$
Root an. cond. $4.19131$
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  + 2·7-s − 3·9-s + 11-s + 4·13-s + 4·17-s − 6·29-s + 2·37-s + 6·41-s − 2·43-s − 3·49-s + 10·53-s + 12·59-s − 6·61-s − 6·63-s + 12·67-s + 16·71-s − 4·73-s + 2·77-s − 4·79-s + 9·81-s − 2·83-s + 6·89-s + 8·91-s + 2·97-s − 3·99-s + 6·101-s − 4·103-s + ⋯
L(s)  = 1  + 0.755·7-s − 9-s + 0.301·11-s + 1.10·13-s + 0.970·17-s − 1.11·29-s + 0.328·37-s + 0.937·41-s − 0.304·43-s − 3/7·49-s + 1.37·53-s + 1.56·59-s − 0.768·61-s − 0.755·63-s + 1.46·67-s + 1.89·71-s − 0.468·73-s + 0.227·77-s − 0.450·79-s + 81-s − 0.219·83-s + 0.635·89-s + 0.838·91-s + 0.203·97-s − 0.301·99-s + 0.597·101-s − 0.394·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(17.5670\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.964527062\)
\(L(\frac12)\) \(\approx\) \(1.964527062\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.937551712026902772018592395716, −8.276577180891466596856050656927, −7.70821531312375135219156255489, −6.66172148738901408428652676215, −5.76358561481875254649875914913, −5.27735578604619471679948891448, −4.07363587216342234630993194198, −3.32473287895092399808930006025, −2.14223797953003720978043765656, −0.956125526926338073634858338162, 0.956125526926338073634858338162, 2.14223797953003720978043765656, 3.32473287895092399808930006025, 4.07363587216342234630993194198, 5.27735578604619471679948891448, 5.76358561481875254649875914913, 6.66172148738901408428652676215, 7.70821531312375135219156255489, 8.276577180891466596856050656927, 8.937551712026902772018592395716

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