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Properties

Label 2-2200-440.107-c0-0-5
Degree $2$
Conductor $2200$
Sign $0.713 - 0.700i$
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 + (0.724 + 0.369i)3-s + (−0.951 − 0.309i)4-s + (−0.478 + 0.658i)6-s + (0.453 − 0.891i)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 + (1.93 − 0.306i)17-s + (0.301 − 0.153i)18-s + (−0.459 − 1.41i)19-s + (0.629 + 0.777i)22-s + (0.658 − 0.478i)24-s + (−0.170 − 1.07i)27-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (0.724 + 0.369i)3-s + (−0.951 − 0.309i)4-s + (−0.478 + 0.658i)6-s + (0.453 − 0.891i)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 + (1.93 − 0.306i)17-s + (0.301 − 0.153i)18-s + (−0.459 − 1.41i)19-s + (0.629 + 0.777i)22-s + (0.658 − 0.478i)24-s + (−0.170 − 1.07i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.713 - 0.700i$
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: \((2,\ 2200,\ (\ :0),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.300426451\)
\(L(\frac12)\) \(\approx\) \(1.300426451\)
\(L(1)\) 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 \( 1 + (-0.669 + 0.743i)T \)
good3 \( 1 + (-0.724 - 0.369i)T + (0.587 + 0.809i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (-1.93 + 0.306i)T + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (0.186 - 0.0949i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.0327 + 0.206i)T + (-0.951 + 0.309i)T^{2} \)
89 \( 1 - 1.82iT - T^{2} \)
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

−9.316706415762202141712308832701, −8.449543737394998873602097276471, −7.999934781506480724925354134865, −7.02006622293272443714887558087, −6.27490606643649447624552526898, −5.50581543150102038182901602195, −4.57477786136827193919083610759, −3.62756194945004792997351231032, −2.92957303534125956493486356310, −1.03441333764093209088052674517, 1.46842521874157592264025124834, 2.13405738839137989503100655015, 3.35032481671131309472893521193, 3.82965365943184757407153702810, 5.04337682237152087018297676946, 5.82501477309541572050866227542, 7.14512370494808915435368794746, 7.87891975879502148903940280563, 8.422851603863591642427848770485, 9.152225505777753498808638362069

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