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Properties

Label 2-943-943.781-c0-0-0
Degree $2$
Conductor $943$
Sign $-0.975 - 0.219i$
Analytic cond. $0.470618$
Root an. cond. $0.686016$
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  + (−1.86 + 0.604i)2-s + (−0.506 + 0.506i)3-s + (2.28 − 1.66i)4-s + (0.636 − 1.24i)6-s + (−2.10 + 2.89i)8-s + 0.486i·9-s + (−0.316 + 2.00i)12-s + (−1.38 − 0.705i)13-s + (1.28 − 3.96i)16-s + (−0.293 − 0.904i)18-s + (0.309 + 0.951i)23-s + (−0.400 − 2.53i)24-s + (−0.309 + 0.951i)25-s + (3.00 + 0.475i)26-s + (−0.753 − 0.753i)27-s + ⋯
L(s)  = 1  + (−1.86 + 0.604i)2-s + (−0.506 + 0.506i)3-s + (2.28 − 1.66i)4-s + (0.636 − 1.24i)6-s + (−2.10 + 2.89i)8-s + 0.486i·9-s + (−0.316 + 2.00i)12-s + (−1.38 − 0.705i)13-s + (1.28 − 3.96i)16-s + (−0.293 − 0.904i)18-s + (0.309 + 0.951i)23-s + (−0.400 − 2.53i)24-s + (−0.309 + 0.951i)25-s + (3.00 + 0.475i)26-s + (−0.753 − 0.753i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(943\)    =    \(23 \cdot 41\)
Sign: $-0.975 - 0.219i$
Analytic conductor: \(0.470618\)
Root analytic conductor: \(0.686016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{943} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 943,\ (\ :0),\ -0.975 - 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2125741432\)
\(L(\frac12)\) \(\approx\) \(0.2125741432\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
good2 \( 1 + (1.86 - 0.604i)T + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.506 - 0.506i)T - iT^{2} \)
5 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
11 \( 1 + (-0.951 + 0.309i)T^{2} \)
13 \( 1 + (1.38 + 0.705i)T + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (0.292 - 1.84i)T + (-0.951 - 0.309i)T^{2} \)
31 \( 1 + (-0.658 - 0.478i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (-0.0475 + 0.0932i)T + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (1.07 - 0.170i)T + (0.951 - 0.309i)T^{2} \)
73 \( 1 - 1.98iT - T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.587 - 0.809i)T^{2} \)
97 \( 1 + (-0.951 - 0.309i)T^{2} \)
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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

−10.35539562738320533368689477586, −9.789870288774525057264829946646, −9.096879961118366754791261926807, −8.101932193396128908316870091106, −7.45141547061495282814828672541, −6.75942282616919671230805687003, −5.48838649286637172713939797654, −5.10542799970754128133918807136, −2.96173965335124078432785729464, −1.62088859145956747480788306046, 0.38354315758073013140926621289, 1.88887943844739407567196683642, 2.82008836860571423314636731758, 4.29287543657237716129239810484, 6.11825804292176088501691098065, 6.75414714454596866419071650628, 7.51108860576183491973801548912, 8.278111183812528481363683314347, 9.212060366901313936482954605284, 9.826493779479039967893125099281

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