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Properties

Label 2-288-96.35-c1-0-11
Degree $2$
Conductor $288$
Sign $0.828 + 0.560i$
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  + (1.32 − 0.495i)2-s + (1.50 − 1.31i)4-s + (−0.0505 + 0.0209i)5-s + (1.44 + 1.44i)7-s + (1.35 − 2.48i)8-s + (−0.0566 + 0.0527i)10-s + (0.320 − 0.132i)11-s + (0.623 − 1.50i)13-s + (2.62 + 1.19i)14-s + (0.559 − 3.96i)16-s − 5.65·17-s + (4.13 + 1.71i)19-s + (−0.0488 + 0.0979i)20-s + (0.359 − 0.335i)22-s + (−3.03 − 3.03i)23-s + ⋯
L(s)  = 1  + (0.936 − 0.350i)2-s + (0.754 − 0.655i)4-s + (−0.0226 + 0.00936i)5-s + (0.544 + 0.544i)7-s + (0.477 − 0.878i)8-s + (−0.0179 + 0.0166i)10-s + (0.0967 − 0.0400i)11-s + (0.172 − 0.417i)13-s + (0.701 + 0.319i)14-s + (0.139 − 0.990i)16-s − 1.37·17-s + (0.947 + 0.392i)19-s + (−0.0109 + 0.0219i)20-s + (0.0766 − 0.0714i)22-s + (−0.632 − 0.632i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.828 + 0.560i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.828 + 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14959 - 0.658372i\)
\(L(\frac12)\) \(\approx\) \(2.14959 - 0.658372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.495i)T \)
3 \( 1 \)
good5 \( 1 + (0.0505 - 0.0209i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.44 - 1.44i)T + 7iT^{2} \)
11 \( 1 + (-0.320 + 0.132i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.623 + 1.50i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + (-4.13 - 1.71i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.03 + 3.03i)T + 23iT^{2} \)
29 \( 1 + (0.721 - 1.74i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 5.26iT - 31T^{2} \)
37 \( 1 + (1.32 + 3.18i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (6.90 - 6.90i)T - 41iT^{2} \)
43 \( 1 + (-3.40 - 8.21i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.23iT - 47T^{2} \)
53 \( 1 + (-0.579 - 1.40i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-4.21 - 10.1i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (12.1 + 5.03i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.34 + 8.08i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-9.36 + 9.36i)T - 71iT^{2} \)
73 \( 1 + (1.72 + 1.72i)T + 73iT^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + (2.48 - 6.00i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.70 - 2.70i)T + 89iT^{2} \)
97 \( 1 - 6.43T + 97T^{2} \)
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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

−11.74928543251863980210446703277, −11.08155340079458619656078389221, −10.09385523145933526691525670319, −8.938793591466452111142609977805, −7.73572320366385130897426004141, −6.53092851978588684422930913318, −5.52063008292216960469365057585, −4.54951259891708036690661612557, −3.22865220335088559499025695689, −1.82157261334651645957301201489, 2.11621120399023762186348735593, 3.79161127657144757478803142742, 4.65412283985516287837187408652, 5.85301526674987305020865606462, 6.94447880369733685176265822566, 7.76465207477052095374602157325, 8.859140276402610156596972974942, 10.22205495969846847122481904483, 11.37583191114663907638523444140, 11.76846876366269955145822089058

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