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Properties

Label 2-24e2-72.59-c1-0-16
Degree $2$
Conductor $576$
Sign $0.985 + 0.166i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  + (1.46 + 0.925i)3-s + (2.06 − 3.57i)5-s + (−0.287 + 0.165i)7-s + (1.28 + 2.71i)9-s + (−3.62 + 2.09i)11-s + (4.27 + 2.46i)13-s + (6.33 − 3.32i)15-s − 3.20i·17-s + 5.42·19-s + (−0.574 − 0.0231i)21-s + (1.39 − 2.41i)23-s + (−6.02 − 10.4i)25-s + (−0.625 + 5.15i)27-s + (−1.03 − 1.79i)29-s + (−3.60 − 2.08i)31-s + ⋯
L(s)  = 1  + (0.845 + 0.534i)3-s + (0.923 − 1.59i)5-s + (−0.108 + 0.0627i)7-s + (0.428 + 0.903i)9-s + (−1.09 + 0.630i)11-s + (1.18 + 0.684i)13-s + (1.63 − 0.858i)15-s − 0.777i·17-s + 1.24·19-s + (−0.125 − 0.00504i)21-s + (0.290 − 0.503i)23-s + (−1.20 − 2.08i)25-s + (−0.120 + 0.992i)27-s + (−0.192 − 0.333i)29-s + (−0.648 − 0.374i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.985 + 0.166i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.985 + 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21213 - 0.185880i\)
\(L(\frac12)\) \(\approx\) \(2.21213 - 0.185880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.46 - 0.925i)T \)
good5 \( 1 + (-2.06 + 3.57i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.287 - 0.165i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.62 - 2.09i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.27 - 2.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.20iT - 17T^{2} \)
19 \( 1 - 5.42T + 19T^{2} \)
23 \( 1 + (-1.39 + 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.03 + 1.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.60 + 2.08i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.21iT - 37T^{2} \)
41 \( 1 + (-1.41 - 0.815i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.99 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.97 + 6.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.05T + 53T^{2} \)
59 \( 1 + (-5.40 - 3.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.10 - 5.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.83 - 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.66T + 71T^{2} \)
73 \( 1 + 6.21T + 73T^{2} \)
79 \( 1 + (0.719 - 0.415i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.96 - 3.44i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.14iT - 89T^{2} \)
97 \( 1 + (0.749 + 1.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.37997130126076941264214982234, −9.540776820767572086605744566855, −9.121666749027205914556043471504, −8.309660554255575972245777218067, −7.38092397880166791943265336327, −5.81044861426216948696536598557, −5.01684243428002233692461393057, −4.22337065671906185583984139570, −2.69411460901683990415591859930, −1.46725340180332465476291078223, 1.68829355981106580424182210613, 3.07953529224249949360339964076, 3.34953147967997937251132169991, 5.59377609977906453287504878528, 6.24170520649619891860071309528, 7.24951189470511511853000924265, 7.937760596463469904410659809826, 8.989268391340651473291483195094, 9.976489307808781384262124573098, 10.64802004520764071266721865383

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