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Properties

Label 2-2061-229.101-c0-0-0
Degree $2$
Conductor $2061$
Sign $-0.983 - 0.183i$
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  + (−0.969 − 0.245i)4-s + (0.0630 + 1.52i)7-s + (−1.87 + 0.393i)13-s + (0.879 + 0.475i)16-s + (−0.740 − 1.13i)19-s + (−0.401 − 0.915i)25-s + (0.313 − 1.49i)28-s + (−0.945 − 0.675i)31-s + (−1.66 + 0.900i)37-s + (−1.08 + 0.584i)43-s + (−1.32 + 0.109i)49-s + (1.91 + 0.0792i)52-s + (1.57 + 1.22i)61-s + (−0.735 − 0.677i)64-s + (0.407 − 0.0507i)67-s + ⋯
L(s)  = 1  + (−0.969 − 0.245i)4-s + (0.0630 + 1.52i)7-s + (−1.87 + 0.393i)13-s + (0.879 + 0.475i)16-s + (−0.740 − 1.13i)19-s + (−0.401 − 0.915i)25-s + (0.313 − 1.49i)28-s + (−0.945 − 0.675i)31-s + (−1.66 + 0.900i)37-s + (−1.08 + 0.584i)43-s + (−1.32 + 0.109i)49-s + (1.91 + 0.0792i)52-s + (1.57 + 1.22i)61-s + (−0.735 − 0.677i)64-s + (0.407 − 0.0507i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2061 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2061\)    =    \(3^{2} \cdot 229\)
Sign: $-0.983 - 0.183i$
Analytic conductor: \(1.02857\)
Root analytic conductor: \(1.01418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2061} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2061,\ (\ :0),\ -0.983 - 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1996344774\)
\(L(\frac12)\) \(\approx\) \(0.1996344774\)
\(L(1)\) 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 \( 1 + (-0.996 + 0.0825i)T \)
good2 \( 1 + (0.969 + 0.245i)T^{2} \)
5 \( 1 + (0.401 + 0.915i)T^{2} \)
7 \( 1 + (-0.0630 - 1.52i)T + (-0.996 + 0.0825i)T^{2} \)
11 \( 1 + (-0.945 + 0.324i)T^{2} \)
13 \( 1 + (1.87 - 0.393i)T + (0.915 - 0.401i)T^{2} \)
17 \( 1 + (-0.401 - 0.915i)T^{2} \)
19 \( 1 + (0.740 + 1.13i)T + (-0.401 + 0.915i)T^{2} \)
23 \( 1 + (0.164 - 0.986i)T^{2} \)
29 \( 1 + (0.996 - 0.0825i)T^{2} \)
31 \( 1 + (0.945 + 0.675i)T + (0.324 + 0.945i)T^{2} \)
37 \( 1 + (1.66 - 0.900i)T + (0.546 - 0.837i)T^{2} \)
41 \( 1 + (-0.969 - 0.245i)T^{2} \)
43 \( 1 + (1.08 - 0.584i)T + (0.546 - 0.837i)T^{2} \)
47 \( 1 + (0.969 - 0.245i)T^{2} \)
53 \( 1 + (0.789 + 0.614i)T^{2} \)
59 \( 1 + (-0.837 + 0.546i)T^{2} \)
61 \( 1 + (-1.57 - 1.22i)T + (0.245 + 0.969i)T^{2} \)
67 \( 1 + (-0.407 + 0.0507i)T + (0.969 - 0.245i)T^{2} \)
71 \( 1 + (-0.945 - 0.324i)T^{2} \)
73 \( 1 + (0.653 - 1.67i)T + (-0.735 - 0.677i)T^{2} \)
79 \( 1 + (0.726 + 0.0300i)T + (0.996 + 0.0825i)T^{2} \)
83 \( 1 + (0.546 + 0.837i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.871 + 0.382i)T + (0.677 + 0.735i)T^{2} \)
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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

−9.769106161506523536740726551680, −8.702337689272647807650112916604, −8.575900727881874249618574610895, −7.36764315491449566763695591830, −6.47736780245490854800500809748, −5.45560653394990487609897513121, −4.99329779170236746387720426768, −4.18469023840372445823130430263, −2.80000247787070962778934225746, −2.01604967372206853150051871032, 0.13771840392495457365039812620, 1.79242889828920834573781842253, 3.39434792059592778235370165561, 3.95182416323686760338592332739, 4.89559603736992598029358906963, 5.46826549180769090758619901829, 6.90179922443769890696151973499, 7.43473003854821815195833733256, 8.072881390540643449034991925855, 8.967437365697923052157751342552

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