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Properties

Label 2-2200-440.179-c0-0-5
Degree $2$
Conductor $2200$
Sign $0.0209 + 0.999i$
Analytic cond. $1.09794$
Root an. cond. $1.04782$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 − 0.809i)2-s + (1.73 − 0.564i)3-s + (−0.309 + 0.951i)4-s + (−1.47 − 1.07i)6-s + (0.951 − 0.309i)8-s + (1.89 − 1.37i)9-s + (0.669 − 0.743i)11-s + 1.82i·12-s + (−0.809 − 0.587i)16-s + (−1.14 + 1.58i)17-s + (−2.22 − 0.722i)18-s + (−0.413 − 1.27i)19-s + (−0.994 − 0.104i)22-s + (1.47 − 1.07i)24-s + (1.43 − 1.97i)27-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (1.73 − 0.564i)3-s + (−0.309 + 0.951i)4-s + (−1.47 − 1.07i)6-s + (0.951 − 0.309i)8-s + (1.89 − 1.37i)9-s + (0.669 − 0.743i)11-s + 1.82i·12-s + (−0.809 − 0.587i)16-s + (−1.14 + 1.58i)17-s + (−2.22 − 0.722i)18-s + (−0.413 − 1.27i)19-s + (−0.994 − 0.104i)22-s + (1.47 − 1.07i)24-s + (1.43 − 1.97i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0209 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0209 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.0209 + 0.999i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.0209 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.585620622\)
\(L(\frac12)\) \(\approx\) \(1.585620622\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 \)
11 \( 1 + (-0.669 + 0.743i)T \)
good3 \( 1 + (-1.73 + 0.564i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (1.14 - 1.58i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.413 + 1.27i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.0646 + 0.198i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + 1.95iT - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.198 - 0.0646i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.122 + 0.169i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + 1.82T + T^{2} \)
97 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
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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

−9.140942765161124827138826527563, −8.375858859087135544886340919659, −7.966268800747880987348712934452, −6.96091624232223728472143399923, −6.34698699341143159957910289873, −4.45508053525852452688256113857, −3.83296315936673648804323270379, −2.96632324800392234407973190304, −2.20369970990147943697058030685, −1.28245586642619164836057351823, 1.72180213610605052072226980103, 2.56022750533935531842784007775, 3.90107370044329172269634422514, 4.45440767500975706481570929323, 5.42445464538055000169370839099, 6.79120268671865031413413926446, 7.21435741413262300843529564170, 8.055799376035768692691979218002, 8.743372931606031490604933358924, 9.195326040594269517060664334392

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