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Properties

Label 2-2061-229.21-c0-0-0
Degree $2$
Conductor $2061$
Sign $0.994 - 0.102i$
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  + (−0.996 − 0.0825i)4-s + (0.490 − 0.292i)7-s + (−0.510 + 1.04i)13-s + (0.986 + 0.164i)16-s + (0.464 + 0.159i)19-s + (0.789 − 0.614i)25-s + (−0.512 + 0.250i)28-s + (0.401 + 0.0842i)31-s + (0.792 − 0.132i)37-s + (1.45 − 0.242i)43-s + (−0.320 + 0.593i)49-s + (0.594 − 0.998i)52-s + (0.644 − 0.700i)61-s + (−0.969 − 0.245i)64-s + (1.79 − 0.0742i)67-s + ⋯
L(s)  = 1  + (−0.996 − 0.0825i)4-s + (0.490 − 0.292i)7-s + (−0.510 + 1.04i)13-s + (0.986 + 0.164i)16-s + (0.464 + 0.159i)19-s + (0.789 − 0.614i)25-s + (−0.512 + 0.250i)28-s + (0.401 + 0.0842i)31-s + (0.792 − 0.132i)37-s + (1.45 − 0.242i)43-s + (−0.320 + 0.593i)49-s + (0.594 − 0.998i)52-s + (0.644 − 0.700i)61-s + (−0.969 − 0.245i)64-s + (1.79 − 0.0742i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2061\)    =    \(3^{2} \cdot 229\)
Sign: $0.994 - 0.102i$
Analytic conductor: \(1.02857\)
Root analytic conductor: \(1.01418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2061} (937, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2061,\ (\ :0),\ 0.994 - 0.102i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9726144265\)
\(L(\frac12)\) \(\approx\) \(0.9726144265\)
\(L(1)\) 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 \( 1 + (0.475 - 0.879i)T \)
good2 \( 1 + (0.996 + 0.0825i)T^{2} \)
5 \( 1 + (-0.789 + 0.614i)T^{2} \)
7 \( 1 + (-0.490 + 0.292i)T + (0.475 - 0.879i)T^{2} \)
11 \( 1 + (0.401 - 0.915i)T^{2} \)
13 \( 1 + (0.510 - 1.04i)T + (-0.614 - 0.789i)T^{2} \)
17 \( 1 + (0.789 - 0.614i)T^{2} \)
19 \( 1 + (-0.464 - 0.159i)T + (0.789 + 0.614i)T^{2} \)
23 \( 1 + (-0.837 - 0.546i)T^{2} \)
29 \( 1 + (-0.475 + 0.879i)T^{2} \)
31 \( 1 + (-0.401 - 0.0842i)T + (0.915 + 0.401i)T^{2} \)
37 \( 1 + (-0.792 + 0.132i)T + (0.945 - 0.324i)T^{2} \)
41 \( 1 + (-0.996 - 0.0825i)T^{2} \)
43 \( 1 + (-1.45 + 0.242i)T + (0.945 - 0.324i)T^{2} \)
47 \( 1 + (0.996 - 0.0825i)T^{2} \)
53 \( 1 + (-0.677 + 0.735i)T^{2} \)
59 \( 1 + (0.324 - 0.945i)T^{2} \)
61 \( 1 + (-0.644 + 0.700i)T + (-0.0825 - 0.996i)T^{2} \)
67 \( 1 + (-1.79 + 0.0742i)T + (0.996 - 0.0825i)T^{2} \)
71 \( 1 + (0.401 + 0.915i)T^{2} \)
73 \( 1 + (0.0899 - 0.721i)T + (-0.969 - 0.245i)T^{2} \)
79 \( 1 + (0.126 - 0.212i)T + (-0.475 - 0.879i)T^{2} \)
83 \( 1 + (0.945 + 0.324i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.202 - 0.259i)T + (-0.245 - 0.969i)T^{2} \)
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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

−9.369959409711197359326921670334, −8.583930290908818734643849918958, −7.88817305983167900640121386244, −7.06972552371498690042160753440, −6.12014624214338854236986732559, −5.11659995789306668036362005176, −4.50659455981172400834104222933, −3.77413724286213304054974203423, −2.46333572574929108221137631303, −1.07989449481880536435659306535, 0.974311554711013269855798877279, 2.57298182029941872256645759865, 3.52429440064876222636034575985, 4.58267821667562172702682933053, 5.21223829433380332618357898534, 5.89559618598243596853867152823, 7.14643426667436789695283946261, 7.910814119217192331395992067222, 8.487939166691956471052820526065, 9.306497747251335817366154862152

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