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Properties

Label 2-24e2-9.2-c2-0-36
Degree $2$
Conductor $576$
Sign $0.507 + 0.861i$
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  + (2.44 − 1.73i)3-s + (4.5 − 2.59i)5-s + (3.17 − 5.49i)7-s + (2.99 − 8.48i)9-s + (8.17 + 4.71i)11-s + (9.84 + 17.0i)13-s + (6.52 − 14.1i)15-s − 1.90i·17-s + 4.69·19-s + (−1.74 − 18.9i)21-s + (−8.17 + 4.71i)23-s + (1 − 1.73i)25-s + (−7.34 − 25.9i)27-s + (2.84 + 1.64i)29-s + (−20.5 − 35.5i)31-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)3-s + (0.900 − 0.519i)5-s + (0.453 − 0.785i)7-s + (0.333 − 0.942i)9-s + (0.743 + 0.429i)11-s + (0.757 + 1.31i)13-s + (0.434 − 0.943i)15-s − 0.112i·17-s + 0.247·19-s + (−0.0832 − 0.903i)21-s + (−0.355 + 0.205i)23-s + (0.0400 − 0.0692i)25-s + (−0.272 − 0.962i)27-s + (0.0982 + 0.0567i)29-s + (−0.662 − 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.507 + 0.861i$
Analytic conductor: \(15.6948\)
Root analytic conductor: \(3.96167\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1),\ 0.507 + 0.861i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.108194553\)
\(L(\frac12)\) \(\approx\) \(3.108194553\)
\(L(2)\) 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 \( 1 + (-2.44 + 1.73i)T \)
good5 \( 1 + (-4.5 + 2.59i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-3.17 + 5.49i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.17 - 4.71i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.84 - 17.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 1.90iT - 289T^{2} \)
19 \( 1 - 4.69T + 361T^{2} \)
23 \( 1 + (8.17 - 4.71i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-2.84 - 1.64i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (20.5 + 35.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 17.3T + 1.36e3T^{2} \)
41 \( 1 + (53.5 - 30.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (0.477 - 0.826i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-12.2 - 7.05i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 9.53iT - 2.80e3T^{2} \)
59 \( 1 + (-79.2 + 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37.5 - 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.4 + 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 85.9iT - 5.04e3T^{2} \)
73 \( 1 + 96.0T + 5.32e3T^{2} \)
79 \( 1 + (-14.8 + 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (76.1 + 43.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 41.3iT - 7.92e3T^{2} \)
97 \( 1 + (47.9 - 83.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.10666254479203801738508596029, −9.316252026094041131753437796972, −8.786639471352460707992263036030, −7.67327168112974257653660060809, −6.85176838554732960505153757443, −5.98653561837124977582850622330, −4.54303079371355204684569781581, −3.66324046590795350632344913595, −1.95189839422647724100599070588, −1.29900846337564612819518336076, 1.69087430835441442655155763358, 2.83094626467658825976903843923, 3.74601650948228487231464284935, 5.23662732531748738294458163012, 5.90584200079107919716061371666, 7.11198830769524451160745770808, 8.479014521004281306714009906800, 8.682299705512713146549368032998, 9.885204847225710673992539414578, 10.43364250958097008457127215803

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