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Properties

Label 2-288-96.59-c1-0-10
Degree $2$
Conductor $288$
Sign $0.954 - 0.296i$
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  + (1.40 + 0.144i)2-s + (1.95 + 0.405i)4-s + (−0.366 + 0.885i)5-s + (1.21 − 1.21i)7-s + (2.69 + 0.853i)8-s + (−0.643 + 1.19i)10-s + (−0.545 + 1.31i)11-s + (0.0270 − 0.0112i)13-s + (1.88 − 1.53i)14-s + (3.67 + 1.58i)16-s − 1.32·17-s + (−1.73 − 4.19i)19-s + (−1.07 + 1.58i)20-s + (−0.956 + 1.77i)22-s + (−0.934 + 0.934i)23-s + ⋯
L(s)  = 1  + (0.994 + 0.101i)2-s + (0.979 + 0.202i)4-s + (−0.164 + 0.396i)5-s + (0.459 − 0.459i)7-s + (0.953 + 0.301i)8-s + (−0.203 + 0.377i)10-s + (−0.164 + 0.396i)11-s + (0.00750 − 0.00310i)13-s + (0.503 − 0.409i)14-s + (0.917 + 0.397i)16-s − 0.321·17-s + (−0.398 − 0.961i)19-s + (−0.241 + 0.354i)20-s + (−0.204 + 0.378i)22-s + (−0.194 + 0.194i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.954 - 0.296i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.954 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28234 + 0.346274i\)
\(L(\frac12)\) \(\approx\) \(2.28234 + 0.346274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.144i)T \)
3 \( 1 \)
good5 \( 1 + (0.366 - 0.885i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.21 + 1.21i)T - 7iT^{2} \)
11 \( 1 + (0.545 - 1.31i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.0270 + 0.0112i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 1.32T + 17T^{2} \)
19 \( 1 + (1.73 + 4.19i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.934 - 0.934i)T - 23iT^{2} \)
29 \( 1 + (9.35 - 3.87i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 9.74iT - 31T^{2} \)
37 \( 1 + (6.28 + 2.60i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.42 + 3.42i)T + 41iT^{2} \)
43 \( 1 + (-0.997 - 0.413i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 6.21iT - 47T^{2} \)
53 \( 1 + (2.94 + 1.22i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-10.4 - 4.32i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.76 - 6.68i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (10.1 - 4.18i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (7.38 + 7.38i)T + 71iT^{2} \)
73 \( 1 + (8.30 - 8.30i)T - 73iT^{2} \)
79 \( 1 - 5.54T + 79T^{2} \)
83 \( 1 + (-11.4 + 4.74i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.93 + 7.93i)T - 89iT^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−11.81290240963625324502676912851, −11.10443000894652940594071838106, −10.37617227851402923966448948023, −8.935050060082619934239555609743, −7.52310696089724411227509642263, −7.03827883696418773105371456585, −5.71871954152362652534800162097, −4.63626277915991009615367868425, −3.58518457739587493606843238356, −2.11258920020439864171156509781, 1.86802288945333373614275127893, 3.40172348277362102340259009546, 4.63842754436141405344301779805, 5.55648633671184492116350524651, 6.59208515610401672322593524631, 7.86383693847907545876461494108, 8.756519132668488188450903783212, 10.18475279799544723234503794886, 11.06188972975176068573651516545, 11.96247310129725216652532606158

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