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Properties

Label 2-2200-440.107-c0-0-1
Degree $2$
Conductor $2200$
Sign $0.455 - 0.890i$
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  + (−0.156 + 0.987i)2-s + (−1.77 − 0.903i)3-s + (−0.951 − 0.309i)4-s + (1.16 − 1.60i)6-s + (0.453 − 0.891i)8-s + (1.73 + 2.39i)9-s + (−0.978 − 0.207i)11-s + (1.40 + 1.40i)12-s + (0.809 + 0.587i)16-s + (−1.32 + 0.209i)17-s + (−2.63 + 1.34i)18-s + (−0.128 − 0.395i)19-s + (0.358 − 0.933i)22-s + (−1.60 + 1.16i)24-s + (−0.608 − 3.84i)27-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−1.77 − 0.903i)3-s + (−0.951 − 0.309i)4-s + (1.16 − 1.60i)6-s + (0.453 − 0.891i)8-s + (1.73 + 2.39i)9-s + (−0.978 − 0.207i)11-s + (1.40 + 1.40i)12-s + (0.809 + 0.587i)16-s + (−1.32 + 0.209i)17-s + (−2.63 + 1.34i)18-s + (−0.128 − 0.395i)19-s + (0.358 − 0.933i)22-s + (−1.60 + 1.16i)24-s + (−0.608 − 3.84i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.455 - 0.890i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.455 - 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3812618800\)
\(L(\frac12)\) \(\approx\) \(0.3812618800\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.156 - 0.987i)T \)
5 \( 1 \)
11 \( 1 + (0.978 + 0.207i)T \)
good3 \( 1 + (1.77 + 0.903i)T + (0.587 + 0.809i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (1.32 - 0.209i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.128 + 0.395i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (-0.773 + 0.251i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + (-1.05 - 1.05i)T + iT^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.62 + 0.829i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.285 - 1.80i)T + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + 0.209iT - T^{2} \)
97 \( 1 + (-1.16 - 0.183i)T + (0.951 + 0.309i)T^{2} \)
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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

−9.311519905980975363319831843059, −8.205591685645572435300960691380, −7.63255374136176772222841431020, −6.87351558224856673375395958125, −6.30854946174687672143988875268, −5.64187924752323057010083609209, −4.93082667920137978171341834140, −4.27283659290116361275049752101, −2.25569909370479821517989178842, −0.818772274130647223016539409803, 0.52703104933626185422003245288, 2.12615785214437752572182958953, 3.53325054838493479965521255340, 4.35860363020198188763172014455, 4.97205679978952550126788189157, 5.64133638431208759672460073754, 6.56209638662194806739226569750, 7.52996427518033079953008169827, 8.708932980703016162350507752092, 9.479194232334310220555351836012

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