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Properties

Label 2-24e2-9.5-c2-0-12
Degree $2$
Conductor $576$
Sign $0.984 - 0.173i$
Analytic cond. $15.6948$
Root an. cond. $3.96167$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)3-s + (−3 − 1.73i)5-s + (1 + 1.73i)7-s + (−4.5 + 7.79i)9-s + (−1.5 + 0.866i)11-s + (−2 + 3.46i)13-s + 10.3i·15-s + 15.5i·17-s + 11·19-s + (3 − 5.19i)21-s + (24 + 13.8i)23-s + (−6.5 − 11.2i)25-s + 27·27-s + (−39 + 22.5i)29-s + (16 − 27.7i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.600 − 0.346i)5-s + (0.142 + 0.247i)7-s + (−0.5 + 0.866i)9-s + (−0.136 + 0.0787i)11-s + (−0.153 + 0.266i)13-s + 0.692i·15-s + 0.916i·17-s + 0.578·19-s + (0.142 − 0.247i)21-s + (1.04 + 0.602i)23-s + (−0.260 − 0.450i)25-s + 27-s + (−1.34 + 0.776i)29-s + (0.516 − 0.893i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.102979642\)
\(L(\frac12)\) \(\approx\) \(1.102979642\)
\(L(2)\) 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 \( 1 + (1.5 + 2.59i)T \)
good5 \( 1 + (3 + 1.73i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 15.5iT - 289T^{2} \)
19 \( 1 - 11T + 361T^{2} \)
23 \( 1 + (-24 - 13.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (39 - 22.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-16 + 27.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 34T + 1.36e3T^{2} \)
41 \( 1 + (10.5 + 6.06i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-30.5 - 52.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42 + 24.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (-43.5 - 25.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-28 - 48.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 + 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 31.1iT - 5.04e3T^{2} \)
73 \( 1 - 65T + 5.32e3T^{2} \)
79 \( 1 + (-19 - 32.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (42 - 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 124. iT - 7.92e3T^{2} \)
97 \( 1 + (-57.5 - 99.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.85198484693850715862652143880, −9.618434741317176943093975983084, −8.572581382397303867333208672477, −7.78957143735692214085182048858, −7.06654235709129802639359049203, −5.94765360546635617755381294071, −5.12685021328306403979954168342, −3.91498812510695906147661681574, −2.37042974589427914833582943205, −1.00575017115617042022763368146, 0.57980085730715615406488364468, 2.85318976357249950557626023844, 3.85285645144225252116132568376, 4.85734618119400710318371752525, 5.71996781703664974399292567417, 6.95394992354790159055253742962, 7.72459354763509428660668275203, 8.939877917741760961263722573382, 9.682108141119806545591372260100, 10.63174223248898877679554641907

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