Properties

Label 2-24e2-576.205-c1-0-71
Degree $2$
Conductor $576$
Sign $-0.837 + 0.547i$
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  + (0.345 − 1.37i)2-s + (−0.245 + 1.71i)3-s + (−1.76 − 0.948i)4-s + (0.286 + 0.326i)5-s + (2.26 + 0.929i)6-s + (−0.859 − 0.113i)7-s + (−1.90 + 2.08i)8-s + (−2.87 − 0.843i)9-s + (0.547 − 0.280i)10-s + (−2.07 − 4.20i)11-s + (2.05 − 2.78i)12-s + (−1.47 − 4.33i)13-s + (−0.452 + 1.13i)14-s + (−0.631 + 0.411i)15-s + (2.20 + 3.33i)16-s + (2.54 + 2.54i)17-s + ⋯
L(s)  = 1  + (0.244 − 0.969i)2-s + (−0.141 + 0.989i)3-s + (−0.880 − 0.474i)4-s + (0.128 + 0.146i)5-s + (0.925 + 0.379i)6-s + (−0.324 − 0.0427i)7-s + (−0.674 + 0.737i)8-s + (−0.959 − 0.281i)9-s + (0.173 − 0.0885i)10-s + (−0.624 − 1.26i)11-s + (0.594 − 0.804i)12-s + (−0.408 − 1.20i)13-s + (−0.120 + 0.304i)14-s + (−0.162 + 0.106i)15-s + (0.550 + 0.834i)16-s + (0.616 + 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.837 + 0.547i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.837 + 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.220000 - 0.738872i\)
\(L(\frac12)\) \(\approx\) \(0.220000 - 0.738872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.345 + 1.37i)T \)
3 \( 1 + (0.245 - 1.71i)T \)
good5 \( 1 + (-0.286 - 0.326i)T + (-0.652 + 4.95i)T^{2} \)
7 \( 1 + (0.859 + 0.113i)T + (6.76 + 1.81i)T^{2} \)
11 \( 1 + (2.07 + 4.20i)T + (-6.69 + 8.72i)T^{2} \)
13 \( 1 + (1.47 + 4.33i)T + (-10.3 + 7.91i)T^{2} \)
17 \( 1 + (-2.54 - 2.54i)T + 17iT^{2} \)
19 \( 1 + (-0.422 + 2.12i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (1.19 + 9.03i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-3.70 + 0.242i)T + (28.7 - 3.78i)T^{2} \)
31 \( 1 + (7.63 - 4.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.03 - 5.19i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (0.367 + 2.78i)T + (-39.6 + 10.6i)T^{2} \)
43 \( 1 + (3.66 - 1.80i)T + (26.1 - 34.1i)T^{2} \)
47 \( 1 + (11.0 - 2.97i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.13 + 4.68i)T + (-20.2 - 48.9i)T^{2} \)
59 \( 1 + (-9.74 - 11.1i)T + (-7.70 + 58.4i)T^{2} \)
61 \( 1 + (-0.568 - 8.66i)T + (-60.4 + 7.96i)T^{2} \)
67 \( 1 + (4.61 + 2.27i)T + (40.7 + 53.1i)T^{2} \)
71 \( 1 + (2.19 + 5.30i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (2.08 - 5.04i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.27 + 12.2i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-7.58 - 6.65i)T + (10.8 + 82.2i)T^{2} \)
89 \( 1 + (-11.7 + 4.87i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (10.7 + 6.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.34197087848895444130672834087, −10.02334407904806476411849266606, −8.671868099086324814323496542262, −8.263090231297850023279802511759, −6.29319528369746171891593649424, −5.47647586969551554789890369410, −4.61973933276952718783606718862, −3.33712637829426569635977078180, −2.77393287410979538465882575815, −0.39925493900069952089762316922, 1.86677539502678981387788186587, 3.48412882804820827773557822228, 4.95234484656160555738724072460, 5.63058046506098169438431836630, 6.79443220262484458442975343872, 7.35849212454819441715043371832, 8.012396596996489261597123441989, 9.362869160970418705462760140561, 9.721322447281456748261325413260, 11.42706133734172838606326513649

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