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Properties

Label 2-288-9.7-c1-0-3
Degree $2$
Conductor $288$
Sign $0.939 - 0.342i$
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  − 1.73·3-s + (0.5 + 0.866i)5-s + (0.866 − 1.5i)7-s + 2.99·9-s + (−0.866 + 1.5i)11-s + (1.5 + 2.59i)13-s + (−0.866 − 1.49i)15-s + 4·17-s + 6.92·19-s + (−1.49 + 2.59i)21-s + (4.33 + 7.5i)23-s + (2 − 3.46i)25-s − 5.19·27-s + (−0.5 + 0.866i)29-s + (−2.59 − 4.5i)31-s + ⋯
L(s)  = 1  − 1.00·3-s + (0.223 + 0.387i)5-s + (0.327 − 0.566i)7-s + 0.999·9-s + (−0.261 + 0.452i)11-s + (0.416 + 0.720i)13-s + (−0.223 − 0.387i)15-s + 0.970·17-s + 1.58·19-s + (−0.327 + 0.566i)21-s + (0.902 + 1.56i)23-s + (0.400 − 0.692i)25-s − 1.00·27-s + (−0.0928 + 0.160i)29-s + (−0.466 − 0.808i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04754 + 0.184710i\)
\(L(\frac12)\) \(\approx\) \(1.04754 + 0.184710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 1.73T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.866 + 1.5i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.866 - 1.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + (-4.33 - 7.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.59 + 4.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.06 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 7.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (-2.59 + 4.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.33 + 7.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + (-1.5 + 2.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.69599145602043291593161112423, −11.02159114446446357568826828766, −10.11739994441904254121245804087, −9.324295476804099714053037445806, −7.59880852691619146494012319413, −7.05762266668823090333220498517, −5.78700925453042974407003725262, −4.90421864001666087939272942155, −3.53905071799655423922912821965, −1.41696961880713688107721738451, 1.15803279730477766421113583115, 3.25279673695194443051173092041, 5.10747130589873339683080282226, 5.43407229370130497301127564280, 6.69708278865877694783776656619, 7.88768642996580561781338385424, 8.942979628153953094322891767711, 10.05471115245081898819961037270, 10.89203574135340061546765166053, 11.77069403583813318423405817064

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