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Properties

Label 2-288-288.11-c1-0-28
Degree $2$
Conductor $288$
Sign $0.257 + 0.966i$
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  + (−1.31 − 0.526i)2-s + (1.34 − 1.09i)3-s + (1.44 + 1.38i)4-s + (0.197 + 1.49i)5-s + (−2.34 + 0.722i)6-s + (−0.749 − 2.79i)7-s + (−1.16 − 2.57i)8-s + (0.620 − 2.93i)9-s + (0.530 − 2.06i)10-s + (−2.13 + 1.63i)11-s + (3.45 + 0.285i)12-s + (3.66 − 4.77i)13-s + (−0.490 + 4.06i)14-s + (1.89 + 1.80i)15-s + (0.173 + 3.99i)16-s + 6.71·17-s + ⋯
L(s)  = 1  + (−0.927 − 0.372i)2-s + (0.776 − 0.629i)3-s + (0.722 + 0.691i)4-s + (0.0881 + 0.669i)5-s + (−0.955 + 0.294i)6-s + (−0.283 − 1.05i)7-s + (−0.412 − 0.910i)8-s + (0.206 − 0.978i)9-s + (0.167 − 0.654i)10-s + (−0.642 + 0.492i)11-s + (0.996 + 0.0824i)12-s + (1.01 − 1.32i)13-s + (−0.131 + 1.08i)14-s + (0.490 + 0.464i)15-s + (0.0434 + 0.999i)16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.257 + 0.966i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869105 - 0.667870i\)
\(L(\frac12)\) \(\approx\) \(0.869105 - 0.667870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.526i)T \)
3 \( 1 + (-1.34 + 1.09i)T \)
good5 \( 1 + (-0.197 - 1.49i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (0.749 + 2.79i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.13 - 1.63i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (-3.66 + 4.77i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 - 6.71T + 17T^{2} \)
19 \( 1 + (-0.341 + 0.141i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.19 + 0.319i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.33 + 0.570i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (2.61 - 1.50i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.12 - 2.71i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.617 - 2.30i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.06 - 5.29i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (7.66 + 4.42i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.18 - 10.0i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-11.3 + 1.49i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (0.560 - 4.26i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (9.36 - 12.2i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (0.817 + 0.817i)T + 71iT^{2} \)
73 \( 1 + (5.70 - 5.70i)T - 73iT^{2} \)
79 \( 1 + (1.89 - 3.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.820 + 0.107i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (4.98 - 4.98i)T - 89iT^{2} \)
97 \( 1 + (-9.22 + 15.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.42633025713131779229569376224, −10.25266249969509080685038623765, −10.07106530225056189690235835602, −8.626099412843452959831485192068, −7.68104807344884751321196003193, −7.25611381117207305377755717215, −6.03630312733077864364186902149, −3.61983844133041354645247162881, −2.88755559392810139894044926530, −1.14354620554282396431618595938, 1.86983581392896782342091093419, 3.37634453574876749530416128640, 5.15978842428525454471388947143, 5.99010079166006760969702048959, 7.53746501178782844356540084494, 8.518528999199163819171646439946, 9.035691556538892305394404820337, 9.739593096548012269834145779629, 10.81995191574169733758244125731, 11.78189324136250967628028766949

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