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Properties

Label 2-2061-229.93-c0-0-0
Degree $2$
Conductor $2061$
Sign $0.505 + 0.862i$
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  + (−0.475 + 0.879i)4-s + (−0.886 − 0.751i)7-s + (0.711 − 1.82i)13-s + (−0.546 − 0.837i)16-s + (0.0663 − 0.151i)19-s + (−0.677 + 0.735i)25-s + (1.08 − 0.422i)28-s + (−0.789 − 0.385i)31-s + (0.863 − 1.32i)37-s + (1.06 − 1.62i)43-s + (0.0577 + 0.345i)49-s + (1.26 + 1.49i)52-s + (−0.0808 − 0.319i)61-s + (0.996 − 0.0825i)64-s + (1.60 + 0.953i)67-s + ⋯
L(s)  = 1  + (−0.475 + 0.879i)4-s + (−0.886 − 0.751i)7-s + (0.711 − 1.82i)13-s + (−0.546 − 0.837i)16-s + (0.0663 − 0.151i)19-s + (−0.677 + 0.735i)25-s + (1.08 − 0.422i)28-s + (−0.789 − 0.385i)31-s + (0.863 − 1.32i)37-s + (1.06 − 1.62i)43-s + (0.0577 + 0.345i)49-s + (1.26 + 1.49i)52-s + (−0.0808 − 0.319i)61-s + (0.996 − 0.0825i)64-s + (1.60 + 0.953i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2061\)    =    \(3^{2} \cdot 229\)
Sign: $0.505 + 0.862i$
Analytic conductor: \(1.02857\)
Root analytic conductor: \(1.01418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2061} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2061,\ (\ :0),\ 0.505 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7898330737\)
\(L(\frac12)\) \(\approx\) \(0.7898330737\)
\(L(1)\) 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 \( 1 + (0.164 + 0.986i)T \)
good2 \( 1 + (0.475 - 0.879i)T^{2} \)
5 \( 1 + (0.677 - 0.735i)T^{2} \)
7 \( 1 + (0.886 + 0.751i)T + (0.164 + 0.986i)T^{2} \)
11 \( 1 + (-0.789 + 0.614i)T^{2} \)
13 \( 1 + (-0.711 + 1.82i)T + (-0.735 - 0.677i)T^{2} \)
17 \( 1 + (-0.677 + 0.735i)T^{2} \)
19 \( 1 + (-0.0663 + 0.151i)T + (-0.677 - 0.735i)T^{2} \)
23 \( 1 + (-0.324 + 0.945i)T^{2} \)
29 \( 1 + (-0.164 - 0.986i)T^{2} \)
31 \( 1 + (0.789 + 0.385i)T + (0.614 + 0.789i)T^{2} \)
37 \( 1 + (-0.863 + 1.32i)T + (-0.401 - 0.915i)T^{2} \)
41 \( 1 + (-0.475 + 0.879i)T^{2} \)
43 \( 1 + (-1.06 + 1.62i)T + (-0.401 - 0.915i)T^{2} \)
47 \( 1 + (0.475 + 0.879i)T^{2} \)
53 \( 1 + (0.245 + 0.969i)T^{2} \)
59 \( 1 + (-0.915 - 0.401i)T^{2} \)
61 \( 1 + (0.0808 + 0.319i)T + (-0.879 + 0.475i)T^{2} \)
67 \( 1 + (-1.60 - 0.953i)T + (0.475 + 0.879i)T^{2} \)
71 \( 1 + (-0.789 - 0.614i)T^{2} \)
73 \( 1 + (-1.98 + 0.0820i)T + (0.996 - 0.0825i)T^{2} \)
79 \( 1 + (1.29 + 1.52i)T + (-0.164 + 0.986i)T^{2} \)
83 \( 1 + (-0.401 + 0.915i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (1.23 - 1.13i)T + (0.0825 - 0.996i)T^{2} \)
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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

−9.201452074047953500798461053646, −8.333048061366893042712537045789, −7.62191808566165164289282006791, −7.09494914677307640540438830191, −5.94864371980256539009874392407, −5.24399674015935092876847480020, −3.84434227895912428492900351412, −3.63823121222463714134562245847, −2.56219545463392595545638016602, −0.60020280988457605384574650968, 1.43884315143555267900194424332, 2.53004254603843337774620963605, 3.83809058935466506615854963321, 4.55890113110714515825977186216, 5.60263318330771807725144045307, 6.32765633773746996799284917470, 6.72108088293962463277454017756, 8.098299483760975182616769924028, 8.880116173979126839060702750002, 9.538836162502793307134188213283

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