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Properties

Label 2-2061-229.141-c0-0-0
Degree $2$
Conductor $2061$
Sign $0.936 + 0.351i$
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  + (−0.164 − 0.986i)4-s + (1.87 + 0.558i)7-s + (1.78 − 0.222i)13-s + (−0.945 + 0.324i)16-s + (−1.38 + 1.08i)19-s + (0.245 + 0.969i)25-s + (0.242 − 1.94i)28-s + (0.677 + 0.264i)31-s + (−1.28 − 0.439i)37-s + (−1.88 − 0.647i)43-s + (2.37 + 1.54i)49-s + (−0.512 − 1.72i)52-s + (0.138 − 1.66i)61-s + (0.475 + 0.879i)64-s + (0.188 − 0.159i)67-s + ⋯
L(s)  = 1  + (−0.164 − 0.986i)4-s + (1.87 + 0.558i)7-s + (1.78 − 0.222i)13-s + (−0.945 + 0.324i)16-s + (−1.38 + 1.08i)19-s + (0.245 + 0.969i)25-s + (0.242 − 1.94i)28-s + (0.677 + 0.264i)31-s + (−1.28 − 0.439i)37-s + (−1.88 − 0.647i)43-s + (2.37 + 1.54i)49-s + (−0.512 − 1.72i)52-s + (0.138 − 1.66i)61-s + (0.475 + 0.879i)64-s + (0.188 − 0.159i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2061\)    =    \(3^{2} \cdot 229\)
Sign: $0.936 + 0.351i$
Analytic conductor: \(1.02857\)
Root analytic conductor: \(1.01418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2061} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2061,\ (\ :0),\ 0.936 + 0.351i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.424849654\)
\(L(\frac12)\) \(\approx\) \(1.424849654\)
\(L(1)\) 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 \( 1 + (0.837 + 0.546i)T \)
good2 \( 1 + (0.164 + 0.986i)T^{2} \)
5 \( 1 + (-0.245 - 0.969i)T^{2} \)
7 \( 1 + (-1.87 - 0.558i)T + (0.837 + 0.546i)T^{2} \)
11 \( 1 + (0.677 - 0.735i)T^{2} \)
13 \( 1 + (-1.78 + 0.222i)T + (0.969 - 0.245i)T^{2} \)
17 \( 1 + (0.245 + 0.969i)T^{2} \)
19 \( 1 + (1.38 - 1.08i)T + (0.245 - 0.969i)T^{2} \)
23 \( 1 + (0.915 + 0.401i)T^{2} \)
29 \( 1 + (-0.837 - 0.546i)T^{2} \)
31 \( 1 + (-0.677 - 0.264i)T + (0.735 + 0.677i)T^{2} \)
37 \( 1 + (1.28 + 0.439i)T + (0.789 + 0.614i)T^{2} \)
41 \( 1 + (-0.164 - 0.986i)T^{2} \)
43 \( 1 + (1.88 + 0.647i)T + (0.789 + 0.614i)T^{2} \)
47 \( 1 + (0.164 - 0.986i)T^{2} \)
53 \( 1 + (-0.0825 + 0.996i)T^{2} \)
59 \( 1 + (-0.614 - 0.789i)T^{2} \)
61 \( 1 + (-0.138 + 1.66i)T + (-0.986 - 0.164i)T^{2} \)
67 \( 1 + (-0.188 + 0.159i)T + (0.164 - 0.986i)T^{2} \)
71 \( 1 + (0.677 + 0.735i)T^{2} \)
73 \( 1 + (0.0710 + 0.0423i)T + (0.475 + 0.879i)T^{2} \)
79 \( 1 + (0.490 + 1.64i)T + (-0.837 + 0.546i)T^{2} \)
83 \( 1 + (0.789 - 0.614i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.629 - 0.159i)T + (0.879 + 0.475i)T^{2} \)
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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

−9.015428285466427546162876440135, −8.513644841949027803232060910384, −8.043688273741356788925778062391, −6.76008301096883576068125482432, −5.93651590670735914103395586946, −5.31971114037079737001338891349, −4.58634635414361141627013608157, −3.60527175517299275860616216369, −1.89965873819847570813165246870, −1.46183842428637967125480211311, 1.36291233465710259737163457310, 2.50244089348088672736857310994, 3.81068533787652798402016073547, 4.38508940205989658060119167557, 5.08502513359852063188053396784, 6.42537174273516385062411903555, 7.05536336961670383017221535077, 8.214689156064178786443286551730, 8.326553650661393390197843735859, 8.929712181879098452447812237481

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