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Properties

Label 2-2200-440.283-c0-0-5
Degree $2$
Conductor $2200$
Sign $0.507 - 0.861i$
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  + (0.987 + 0.156i)2-s + (−0.369 + 0.724i)3-s + (0.951 + 0.309i)4-s + (−0.478 + 0.658i)6-s + (0.891 + 0.453i)8-s + (0.198 + 0.273i)9-s + (0.669 − 0.743i)11-s + (−0.575 + 0.575i)12-s + (0.809 + 0.587i)16-s + (−0.306 − 1.93i)17-s + (0.153 + 0.301i)18-s + (0.459 + 1.41i)19-s + (0.777 − 0.629i)22-s + (−0.658 + 0.478i)24-s + (−1.07 + 0.170i)27-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (−0.369 + 0.724i)3-s + (0.951 + 0.309i)4-s + (−0.478 + 0.658i)6-s + (0.891 + 0.453i)8-s + (0.198 + 0.273i)9-s + (0.669 − 0.743i)11-s + (−0.575 + 0.575i)12-s + (0.809 + 0.587i)16-s + (−0.306 − 1.93i)17-s + (0.153 + 0.301i)18-s + (0.459 + 1.41i)19-s + (0.777 − 0.629i)22-s + (−0.658 + 0.478i)24-s + (−1.07 + 0.170i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.507 - 0.861i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (2043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.507 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.103067939\)
\(L(\frac12)\) \(\approx\) \(2.103067939\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 - 0.156i)T \)
5 \( 1 \)
11 \( 1 + (-0.669 + 0.743i)T \)
good3 \( 1 + (0.369 - 0.724i)T + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.306 + 1.93i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (-0.459 - 1.41i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (1.89 - 0.614i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (1.14 - 1.14i)T - iT^{2} \)
47 \( 1 + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.294 + 0.294i)T - iT^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.0949 + 0.186i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.206 + 0.0327i)T + (0.951 - 0.309i)T^{2} \)
89 \( 1 + 1.82iT - T^{2} \)
97 \( 1 + (-0.183 + 1.16i)T + (-0.951 - 0.309i)T^{2} \)
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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

−9.596100984823065136719269503680, −8.498553448015191312158840545629, −7.65532355089768926790906357261, −6.86210082864158713584634708222, −6.05215456611540188734418521014, −5.20812053425233017580976716259, −4.70866131152193831431945511867, −3.72751575438736658644606062229, −3.03809515313869877718793999762, −1.65643491397665749721638554500, 1.35722236123161300246844486332, 2.15065066156875535538785862560, 3.50495727409940926486775339518, 4.20959519919483093561222783247, 5.14566602291759791229649804231, 6.04150587524212431681522633388, 6.80133009580387185329173611585, 7.06077471860965874306618192203, 8.156017217697419791769307544053, 9.155111045424394759402555112624

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