Properties

Label 2-288-288.275-c1-0-10
Degree $2$
Conductor $288$
Sign $0.801 - 0.597i$
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  + (−1.17 − 0.785i)2-s + (−1.01 − 1.40i)3-s + (0.764 + 1.84i)4-s + (2.85 + 0.376i)5-s + (0.0905 + 2.44i)6-s + (−0.838 + 3.13i)7-s + (0.553 − 2.77i)8-s + (−0.939 + 2.84i)9-s + (−3.06 − 2.68i)10-s + (−3.69 + 4.81i)11-s + (1.81 − 2.94i)12-s + (−3.49 + 2.68i)13-s + (3.44 − 3.02i)14-s + (−2.37 − 4.39i)15-s + (−2.83 + 2.82i)16-s + 4.67·17-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)2-s + (−0.586 − 0.810i)3-s + (0.382 + 0.924i)4-s + (1.27 + 0.168i)5-s + (0.0369 + 0.999i)6-s + (−0.317 + 1.18i)7-s + (0.195 − 0.980i)8-s + (−0.313 + 0.949i)9-s + (−0.968 − 0.849i)10-s + (−1.11 + 1.45i)11-s + (0.524 − 0.851i)12-s + (−0.970 + 0.744i)13-s + (0.921 − 0.807i)14-s + (−0.612 − 1.13i)15-s + (−0.707 + 0.706i)16-s + 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.801 - 0.597i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.801 - 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649421 + 0.215467i\)
\(L(\frac12)\) \(\approx\) \(0.649421 + 0.215467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.785i)T \)
3 \( 1 + (1.01 + 1.40i)T \)
good5 \( 1 + (-2.85 - 0.376i)T + (4.82 + 1.29i)T^{2} \)
7 \( 1 + (0.838 - 3.13i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.69 - 4.81i)T + (-2.84 - 10.6i)T^{2} \)
13 \( 1 + (3.49 - 2.68i)T + (3.36 - 12.5i)T^{2} \)
17 \( 1 - 4.67T + 17T^{2} \)
19 \( 1 + (1.17 - 2.83i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.57 + 0.957i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.568 + 4.31i)T + (-28.0 + 7.50i)T^{2} \)
31 \( 1 + (3.52 + 2.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.98 + 0.820i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.0956 - 0.357i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.627 - 0.481i)T + (11.1 + 41.5i)T^{2} \)
47 \( 1 + (-2.98 + 1.72i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.59 - 2.73i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.174 + 1.32i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (2.52 - 0.331i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (-8.04 + 6.17i)T + (17.3 - 64.7i)T^{2} \)
71 \( 1 + (-5.75 + 5.75i)T - 71iT^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 + (-7.01 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.31 + 17.6i)T + (-80.1 + 21.4i)T^{2} \)
89 \( 1 + (-2.22 - 2.22i)T + 89iT^{2} \)
97 \( 1 + (-0.970 - 1.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.17397511160613693536499950387, −10.86791978940493404695766506248, −9.847302482268427260345743075800, −9.476533904911375038032125038069, −8.015761841017350197482788638845, −7.14388101632812149469492555496, −6.07623964830356555891312912019, −5.06887096277029933373717546659, −2.48679755764171339769438507448, −1.99220987431463163153813125628, 0.69660154553572047428425693000, 3.05893605330352143802147444969, 5.15650981968217950173977646162, 5.59266555466031004213371553355, 6.73607425652437579666956072247, 7.88404252608594341429855585716, 9.138420907755777101284888360650, 9.911453912942338380356566173648, 10.54490017556726338968673191438, 11.04577271959173000709721263294

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