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Properties

Label 2-2061-229.177-c0-0-0
Degree $2$
Conductor $2061$
Sign $0.705 + 0.708i$
Analytic cond. $1.02857$
Root an. cond. $1.01418$
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  + (−0.614 − 0.789i)4-s + (0.0769 + 0.0300i)7-s + (0.981 + 0.292i)13-s + (−0.245 + 0.969i)16-s + (0.706 + 0.382i)19-s + (0.546 − 0.837i)25-s + (−0.0235 − 0.0792i)28-s + (0.986 − 1.16i)31-s + (−0.484 − 1.91i)37-s + (0.159 + 0.629i)43-s + (−0.730 − 0.672i)49-s + (−0.372 − 0.953i)52-s + (−1.39 − 0.477i)61-s + (0.915 − 0.401i)64-s + (0.512 − 0.250i)67-s + ⋯
L(s)  = 1  + (−0.614 − 0.789i)4-s + (0.0769 + 0.0300i)7-s + (0.981 + 0.292i)13-s + (−0.245 + 0.969i)16-s + (0.706 + 0.382i)19-s + (0.546 − 0.837i)25-s + (−0.0235 − 0.0792i)28-s + (0.986 − 1.16i)31-s + (−0.484 − 1.91i)37-s + (0.159 + 0.629i)43-s + (−0.730 − 0.672i)49-s + (−0.372 − 0.953i)52-s + (−1.39 − 0.477i)61-s + (0.915 − 0.401i)64-s + (0.512 − 0.250i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2061\)    =    \(3^{2} \cdot 229\)
Sign: $0.705 + 0.708i$
Analytic conductor: \(1.02857\)
Root analytic conductor: \(1.01418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2061} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2061,\ (\ :0),\ 0.705 + 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060452503\)
\(L(\frac12)\) \(\approx\) \(1.060452503\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
229 \( 1 + (0.735 + 0.677i)T \)
good2 \( 1 + (0.614 + 0.789i)T^{2} \)
5 \( 1 + (-0.546 + 0.837i)T^{2} \)
7 \( 1 + (-0.0769 - 0.0300i)T + (0.735 + 0.677i)T^{2} \)
11 \( 1 + (0.986 - 0.164i)T^{2} \)
13 \( 1 + (-0.981 - 0.292i)T + (0.837 + 0.546i)T^{2} \)
17 \( 1 + (0.546 - 0.837i)T^{2} \)
19 \( 1 + (-0.706 - 0.382i)T + (0.546 + 0.837i)T^{2} \)
23 \( 1 + (-0.996 - 0.0825i)T^{2} \)
29 \( 1 + (-0.735 - 0.677i)T^{2} \)
31 \( 1 + (-0.986 + 1.16i)T + (-0.164 - 0.986i)T^{2} \)
37 \( 1 + (0.484 + 1.91i)T + (-0.879 + 0.475i)T^{2} \)
41 \( 1 + (-0.614 - 0.789i)T^{2} \)
43 \( 1 + (-0.159 - 0.629i)T + (-0.879 + 0.475i)T^{2} \)
47 \( 1 + (0.614 - 0.789i)T^{2} \)
53 \( 1 + (0.945 + 0.324i)T^{2} \)
59 \( 1 + (0.475 - 0.879i)T^{2} \)
61 \( 1 + (1.39 + 0.477i)T + (0.789 + 0.614i)T^{2} \)
67 \( 1 + (-0.512 + 0.250i)T + (0.614 - 0.789i)T^{2} \)
71 \( 1 + (0.986 + 0.164i)T^{2} \)
73 \( 1 + (-1.59 + 0.334i)T + (0.915 - 0.401i)T^{2} \)
79 \( 1 + (-0.711 - 1.82i)T + (-0.735 + 0.677i)T^{2} \)
83 \( 1 + (-0.879 - 0.475i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (1.62 - 1.06i)T + (0.401 - 0.915i)T^{2} \)
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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

−9.324677516593207663788380522171, −8.495176827509102513198757763459, −7.84476346303341139680765399295, −6.65627952986479550338022312281, −6.03149126804027703935120545044, −5.24724995762836450013459628976, −4.36828486555592519040403113333, −3.56930294240008492587227167776, −2.16470389066727843652982616073, −0.955127292869200122622876721431, 1.25942301374903281258607465362, 2.95884584301470603633934984340, 3.47884118252817192165470821090, 4.60367509535479472943173193013, 5.21951415061908316790043944694, 6.35439954548719790968838416283, 7.16141447257839856059612313622, 8.002934709869528584476099360380, 8.605870827615751940638000106344, 9.232159599135928982118932738209

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