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Properties

Label 2-288-32.13-c1-0-2
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

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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: $\chi_{288} (109, \cdot )$
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 \( 1 + 3.30T + 97T^{2} \)
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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

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

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