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Properties

Label 2-2200-440.307-c0-0-4
Degree $2$
Conductor $2200$
Sign $0.229 + 0.973i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)8-s + i·9-s + 11-s − 1.00·16-s + (1.41 − 1.41i)17-s + (0.707 + 0.707i)18-s + (0.707 − 0.707i)22-s + (−0.707 + 0.707i)32-s − 2.00i·34-s + 1.00·36-s + (−1.41 − 1.41i)43-s − 1.00i·44-s − i·49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)8-s + i·9-s + 11-s − 1.00·16-s + (1.41 − 1.41i)17-s + (0.707 + 0.707i)18-s + (0.707 − 0.707i)22-s + (−0.707 + 0.707i)32-s − 2.00i·34-s + 1.00·36-s + (−1.41 − 1.41i)43-s − 1.00i·44-s − i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.791086100\)
\(L(\frac12)\) \(\approx\) \(1.791086100\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - iT^{2} \)
7 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.299228752317045225674208886739, −8.422061285150216218169422115553, −7.35842682848942848428173866362, −6.70748462063807743199287554953, −5.55492931092716839074808250970, −5.12768064960214409610082790981, −4.13540349568350639403692320929, −3.27442882769064196710997742391, −2.31954011782605909038660700931, −1.19480678919606056846983396169, 1.52172224152395860185215973145, 3.18032445802938551575537015275, 3.70289616598788391295310560450, 4.55896761896784040230187923818, 5.64403088764313433412672335784, 6.28968439828387738384192254602, 6.81167133630004105760389580499, 7.86310975529565528852622734310, 8.410362336068708695013612630629, 9.318449046558392797248226787404

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