↑

Properties

Label 2-1044-116.71-c0-0-1
Degree $2$
Conductor $1044$
Sign $0.617 + 0.786i$
Analytic cond. $0.521023$
Root an. cond. $0.721819$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (1.75 + 0.846i)5-s + (−0.974 + 0.222i)8-s + (1.52 − 1.21i)10-s + (−0.277 + 1.21i)13-s + (−0.222 + 0.974i)16-s − 1.80i·17-s + (−0.433 − 1.90i)20-s + (1.74 + 2.19i)25-s + (0.974 + 0.777i)26-s + (−0.974 − 0.222i)29-s + (0.781 + 0.623i)32-s + (−1.62 − 0.781i)34-s + (−1.52 + 0.347i)37-s + ⋯
L(s)  = 1  + (0.433 − 0.900i)2-s + (−0.623 − 0.781i)4-s + (1.75 + 0.846i)5-s + (−0.974 + 0.222i)8-s + (1.52 − 1.21i)10-s + (−0.277 + 1.21i)13-s + (−0.222 + 0.974i)16-s − 1.80i·17-s + (−0.433 − 1.90i)20-s + (1.74 + 2.19i)25-s + (0.974 + 0.777i)26-s + (−0.974 − 0.222i)29-s + (0.781 + 0.623i)32-s + (−1.62 − 0.781i)34-s + (−1.52 + 0.347i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1044\)    =    \(2^{2} \cdot 3^{2} \cdot 29\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(0.521023\)
Root analytic conductor: \(0.721819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1044} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1044,\ (\ :0),\ 0.617 + 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.503160389\)
\(L(\frac12)\) \(\approx\) \(1.503160389\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.433 + 0.900i)T \)
3 \( 1 \)
29 \( 1 + (0.974 + 0.222i)T \)
good5 \( 1 + (-1.75 - 0.846i)T + (0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + 1.80iT - T^{2} \)
19 \( 1 + (-0.222 + 0.974i)T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (1.52 - 0.347i)T + (0.900 - 0.433i)T^{2} \)
41 \( 1 + 1.24iT - T^{2} \)
43 \( 1 + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (0.781 + 0.376i)T + (0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + (-0.900 + 0.433i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.193 + 0.400i)T + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (1.22 - 0.974i)T + (0.222 - 0.974i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.965173604961707005283472597297, −9.467580794553630623586831024111, −8.919660642805891227873924198682, −7.13077524449753602973849276440, −6.53590799111415027172865116431, −5.51256349772996308307043620987, −4.90567290349856394736385933568, −3.48034749644108209858828074438, −2.45841492275173339965052598435, −1.77051201290097105919044043809, 1.66822713734094875923289264307, 3.08157601856144236119350673143, 4.41886609573478985781041596688, 5.39042178325603018177986586936, 5.83025456121361463707323022826, 6.56969450525462072010758910908, 7.82170179278889915698557416197, 8.542520491530779369708725278994, 9.271516029669891407721781355387, 10.03430655595909552257332349462

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