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Properties

Label 2-24e2-144.11-c1-0-4
Degree $2$
Conductor $576$
Sign $0.998 + 0.0517i$
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  + (−1.61 − 0.632i)3-s + (−3.81 − 1.02i)5-s + (−1.46 + 2.54i)7-s + (2.20 + 2.03i)9-s + (2.65 − 0.710i)11-s + (−2.34 − 0.628i)13-s + (5.50 + 4.05i)15-s − 2.89i·17-s + (1.99 + 1.99i)19-s + (3.97 − 3.17i)21-s + (2.07 − 1.19i)23-s + (9.17 + 5.29i)25-s + (−2.26 − 4.67i)27-s + (8.46 − 2.26i)29-s + (0.439 − 0.253i)31-s + ⋯
L(s)  = 1  + (−0.931 − 0.364i)3-s + (−1.70 − 0.457i)5-s + (−0.554 + 0.960i)7-s + (0.733 + 0.679i)9-s + (0.799 − 0.214i)11-s + (−0.651 − 0.174i)13-s + (1.42 + 1.04i)15-s − 0.702i·17-s + (0.458 + 0.458i)19-s + (0.866 − 0.691i)21-s + (0.431 − 0.249i)23-s + (1.83 + 1.05i)25-s + (−0.435 − 0.900i)27-s + (1.57 − 0.421i)29-s + (0.0788 − 0.0455i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.998 + 0.0517i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.998 + 0.0517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653948 - 0.0169387i\)
\(L(\frac12)\) \(\approx\) \(0.653948 - 0.0169387i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 0.632i)T \)
good5 \( 1 + (3.81 + 1.02i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.46 - 2.54i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.65 + 0.710i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.34 + 0.628i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.89iT - 17T^{2} \)
19 \( 1 + (-1.99 - 1.99i)T + 19iT^{2} \)
23 \( 1 + (-2.07 + 1.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.46 + 2.26i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.439 + 0.253i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.36 - 1.36i)T + 37iT^{2} \)
41 \( 1 + (-0.745 - 1.29i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.27 - 4.74i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.25 + 5.64i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.17 - 5.17i)T - 53iT^{2} \)
59 \( 1 + (0.664 - 2.48i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.99 - 11.1i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.53 + 9.46i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.65iT - 71T^{2} \)
73 \( 1 + 4.91iT - 73T^{2} \)
79 \( 1 + (-3.61 - 2.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.37 - 12.5i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 7.33T + 89T^{2} \)
97 \( 1 + (-2.50 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.05803416576780698026849908651, −9.857670400023843939658668087429, −8.862470200744506414319978029323, −7.952604688347036894249989290101, −7.13363837765114858456232677202, −6.21560733806150757546786548019, −5.08433904000545243757465926462, −4.25552036770229042160199565036, −2.91571992586571262691382449192, −0.793990271637767171005404029633, 0.69669374719075659252168327567, 3.35405443844478768682645068721, 4.08113711774450087389646503417, 4.86599363741434226276511646305, 6.51756648468385065222156058936, 6.99677034494592435541282236963, 7.80303028336284076089448559446, 9.096147229017485458681996912935, 10.14652963819185644515643180512, 10.79891617760818676783795265649

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