↑

Properties

Label 2-288-32.5-c1-0-10
Degree $2$
Conductor $288$
Sign $0.902 + 0.430i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.605 + 1.27i)2-s + (−1.26 − 1.54i)4-s + (−1.60 − 0.666i)5-s + (−0.589 − 0.589i)7-s + (2.74 − 0.681i)8-s + (1.82 − 1.65i)10-s + (0.657 − 1.58i)11-s + (3.87 − 1.60i)13-s + (1.11 − 0.396i)14-s + (−0.791 + 3.92i)16-s − 7.96i·17-s + (3.97 − 1.64i)19-s + (1.00 + 3.33i)20-s + (1.62 + 1.80i)22-s + (0.452 − 0.452i)23-s + ⋯
L(s)  = 1  + (−0.428 + 0.903i)2-s + (−0.633 − 0.773i)4-s + (−0.719 − 0.298i)5-s + (−0.222 − 0.222i)7-s + (0.970 − 0.240i)8-s + (0.577 − 0.522i)10-s + (0.198 − 0.478i)11-s + (1.07 − 0.444i)13-s + (0.296 − 0.105i)14-s + (−0.197 + 0.980i)16-s − 1.93i·17-s + (0.911 − 0.377i)19-s + (0.225 + 0.745i)20-s + (0.347 + 0.383i)22-s + (0.0942 − 0.0942i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767473 - 0.173781i\)
\(L(\frac12)\) \(\approx\) \(0.767473 - 0.173781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.605 - 1.27i)T \)
3 \( 1 \)
good5 \( 1 + (1.60 + 0.666i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.589 + 0.589i)T + 7iT^{2} \)
11 \( 1 + (-0.657 + 1.58i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-3.87 + 1.60i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 7.96iT - 17T^{2} \)
19 \( 1 + (-3.97 + 1.64i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.452 + 0.452i)T - 23iT^{2} \)
29 \( 1 + (-1.69 - 4.08i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 9.32T + 31T^{2} \)
37 \( 1 + (-0.810 - 0.335i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.65 + 6.65i)T - 41iT^{2} \)
43 \( 1 + (2.22 - 5.36i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.50iT - 47T^{2} \)
53 \( 1 + (1.10 - 2.67i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.92 - 0.796i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.70 + 11.3i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-4.54 - 10.9i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.09 + 9.09i)T + 71iT^{2} \)
73 \( 1 + (-1.65 + 1.65i)T - 73iT^{2} \)
79 \( 1 + 0.580iT - 79T^{2} \)
83 \( 1 + (3.33 - 1.38i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.91 - 4.91i)T + 89iT^{2} \)
97 \( 1 + 3.30T + 97T^{2} \)
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

−11.55148204771097267568414599788, −10.80057177557253506991243641108, −9.507965381874212726012911066442, −8.843356072507640730474686095573, −7.78064695035763845689746508335, −7.06175044800925534046201181979, −5.83499719802132688910422622210, −4.78603516633842416773719370272, −3.44962693560248032831835855671, −0.74441934947728141005774065963, 1.70075158833675162526671330421, 3.45651758945487037150254274078, 4.14477157728851043333501541354, 5.88887738529291840389011231924, 7.30398965124236912063604469972, 8.218644144383912451245885986872, 9.105966891693699744732846561860, 10.12145802730501208393119128787, 11.03342623231076319570419162046, 11.70453255371662402038695853854

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