Properties

Label 2-24e2-16.5-c1-0-4
Degree $2$
Conductor $576$
Sign $0.807 + 0.589i$
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  + (−1.74 − 1.74i)5-s + 2.55i·7-s + (0.473 + 0.473i)11-s + (2.88 − 2.88i)13-s + 6.44·17-s + (4.55 − 4.55i)19-s − 2.82i·23-s + 1.11i·25-s + (3.07 − 3.07i)29-s − 6.55·31-s + (4.47 − 4.47i)35-s + (−2.72 − 2.72i)37-s + 0.788i·41-s + (0.389 + 0.389i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (−0.782 − 0.782i)5-s + 0.966i·7-s + (0.142 + 0.142i)11-s + (0.800 − 0.800i)13-s + 1.56·17-s + (1.04 − 1.04i)19-s − 0.589i·23-s + 0.223i·25-s + (0.571 − 0.571i)29-s − 1.17·31-s + (0.756 − 0.756i)35-s + (−0.448 − 0.448i)37-s + 0.123i·41-s + (0.0594 + 0.0594i)43-s + 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.807 + 0.589i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.807 + 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27475 - 0.415427i\)
\(L(\frac12)\) \(\approx\) \(1.27475 - 0.415427i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.74 + 1.74i)T + 5iT^{2} \)
7 \( 1 - 2.55iT - 7T^{2} \)
11 \( 1 + (-0.473 - 0.473i)T + 11iT^{2} \)
13 \( 1 + (-2.88 + 2.88i)T - 13iT^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 + (-4.55 + 4.55i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-3.07 + 3.07i)T - 29iT^{2} \)
31 \( 1 + 6.55T + 31T^{2} \)
37 \( 1 + (2.72 + 2.72i)T + 37iT^{2} \)
41 \( 1 - 0.788iT - 41T^{2} \)
43 \( 1 + (-0.389 - 0.389i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (-2.57 - 2.57i)T + 53iT^{2} \)
59 \( 1 + (-4 - 4i)T + 59iT^{2} \)
61 \( 1 + (4.38 - 4.38i)T - 61iT^{2} \)
67 \( 1 + (-2.11 + 2.11i)T - 67iT^{2} \)
71 \( 1 - 5.11iT - 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 - 6.31T + 79T^{2} \)
83 \( 1 + (-0.641 + 0.641i)T - 83iT^{2} \)
89 \( 1 - 6.31iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−10.70281473188622983597193902894, −9.619305555263577973139029124799, −8.765536065123437006138588589961, −8.128998718856790687488967019441, −7.22424661365139226745822892639, −5.80247040770925080392553328473, −5.18354418331585345340070593340, −3.93369867784828193887568768757, −2.80948793012491541233856096576, −0.941512139892790684490703921037, 1.33156430944238143679029433184, 3.48306481448989052587769734648, 3.70226107865529952432717930047, 5.26862656594871581539256273850, 6.43660008709467037697971402029, 7.41104250492814925896222859909, 7.82780130399494396146908052753, 9.103914861881825241255724181386, 10.13139352365769121363766088341, 10.78764105106522216393675781024

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