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Properties

Label 2-943-943.459-c0-0-1
Degree $2$
Conductor $943$
Sign $-0.164 - 0.986i$
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  + (−0.122 + 0.169i)2-s + (1.18 + 1.18i)3-s + (0.295 + 0.909i)4-s + (−0.346 + 0.0548i)6-s + (−0.388 − 0.126i)8-s + 1.81i·9-s + (−0.728 + 1.42i)12-s + (−0.312 − 1.97i)13-s + (−0.704 + 0.511i)16-s + (−0.306 − 0.222i)18-s + (−0.809 − 0.587i)23-s + (−0.311 − 0.611i)24-s + (0.809 − 0.587i)25-s + (0.372 + 0.189i)26-s + (−0.964 + 0.964i)27-s + ⋯
L(s)  = 1  + (−0.122 + 0.169i)2-s + (1.18 + 1.18i)3-s + (0.295 + 0.909i)4-s + (−0.346 + 0.0548i)6-s + (−0.388 − 0.126i)8-s + 1.81i·9-s + (−0.728 + 1.42i)12-s + (−0.312 − 1.97i)13-s + (−0.704 + 0.511i)16-s + (−0.306 − 0.222i)18-s + (−0.809 − 0.587i)23-s + (−0.311 − 0.611i)24-s + (0.809 − 0.587i)25-s + (0.372 + 0.189i)26-s + (−0.964 + 0.964i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(943\)    =    \(23 \cdot 41\)
Sign: $-0.164 - 0.986i$
Analytic conductor: \(0.470618\)
Root analytic conductor: \(0.686016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{943} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 943,\ (\ :0),\ -0.164 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.398373423\)
\(L(\frac12)\) \(\approx\) \(1.398373423\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.104 - 0.994i)T \)
good2 \( 1 + (0.122 - 0.169i)T + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (-1.18 - 1.18i)T + iT^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.951 - 0.309i)T^{2} \)
11 \( 1 + (-0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.312 + 1.97i)T + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.494 + 0.970i)T + (-0.587 - 0.809i)T^{2} \)
31 \( 1 + (0.128 - 0.395i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (1.84 - 0.292i)T + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (1.12 - 0.571i)T + (0.587 - 0.809i)T^{2} \)
73 \( 1 - 1.48iT - T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.951 - 0.309i)T^{2} \)
97 \( 1 + (-0.587 - 0.809i)T^{2} \)
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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

−10.23338261488597586079848796492, −9.693998295244531429741192136485, −8.535739519029354894441448441796, −8.243682541563528247789762057203, −7.52835214628119341080994062671, −6.25159256925426150641462049551, −4.96201428547072363989590327205, −4.06820306395543235211686686591, −3.08560173574850587271166400301, −2.59844099779268342328565780217, 1.53451145443997841494593188144, 2.12042993710964606406707275351, 3.31792285121237940632647144444, 4.67715410582025263938431853010, 6.00173345082497117113541738170, 6.89856057544327899773352480485, 7.27071466461118346272426713189, 8.510212946613544092881693217068, 9.129251401243376416667396488822, 9.742056443472887969913605893754

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