↑

Properties

Label 2-544-17.5-c0-0-0
Degree $2$
Conductor $544$
Sign $0.813 - 0.581i$
Analytic cond. $0.271491$
Root an. cond. $0.521048$
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  + (−0.216 + 1.08i)5-s + (0.382 − 0.923i)9-s + (1.30 + 1.30i)13-s + (−0.707 + 0.707i)17-s + (−0.216 − 0.0897i)25-s + (−0.382 − 0.0761i)29-s + (−1.08 − 1.63i)37-s + (−0.382 − 1.92i)41-s + (0.923 + 0.617i)45-s + (0.923 − 0.382i)49-s + (−0.292 − 0.707i)53-s + (−1.63 + 0.324i)61-s + (−1.70 + 1.14i)65-s + (−0.0761 + 0.382i)73-s + (−0.707 − 0.707i)81-s + ⋯
L(s)  = 1  + (−0.216 + 1.08i)5-s + (0.382 − 0.923i)9-s + (1.30 + 1.30i)13-s + (−0.707 + 0.707i)17-s + (−0.216 − 0.0897i)25-s + (−0.382 − 0.0761i)29-s + (−1.08 − 1.63i)37-s + (−0.382 − 1.92i)41-s + (0.923 + 0.617i)45-s + (0.923 − 0.382i)49-s + (−0.292 − 0.707i)53-s + (−1.63 + 0.324i)61-s + (−1.70 + 1.14i)65-s + (−0.0761 + 0.382i)73-s + (−0.707 − 0.707i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.813 - 0.581i$
Analytic conductor: \(0.271491\)
Root analytic conductor: \(0.521048\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :0),\ 0.813 - 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9013605501\)
\(L(\frac12)\) \(\approx\) \(0.9013605501\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
97 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
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

−11.00054652638078806627534287014, −10.46682142078004686956244416056, −9.218390969785557518798186454966, −8.657286844241876680900412660539, −7.20510575885949919596270229869, −6.70088631359301043378855550314, −5.83844298349978782689363244746, −4.10520438691444965839565846077, −3.53372728561153302885629137637, −1.91911946694347466477349107245, 1.36128957131794691255204997358, 3.08755474479445968235233227101, 4.48998568235527809037970386754, 5.16748239160056455420473703524, 6.27239462408285122327252201662, 7.58032761229907992909239210462, 8.318931835025816229912597778668, 8.983833330203497868127654990959, 10.14334993604392975719401580736, 10.89914398874432347050582510547

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