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Properties

Label 2-24e2-144.85-c1-0-1
Degree $2$
Conductor $576$
Sign $-0.978 + 0.206i$
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  + (−0.841 + 1.51i)3-s + (−0.0691 + 0.0185i)5-s + (−1.28 + 0.740i)7-s + (−1.58 − 2.54i)9-s + (−0.587 + 2.19i)11-s + (0.104 + 0.388i)13-s + (0.0301 − 0.120i)15-s − 0.851·17-s + (−3.75 + 3.75i)19-s + (−0.0420 − 2.56i)21-s + (−7.44 − 4.29i)23-s + (−4.32 + 2.49i)25-s + (5.18 − 0.255i)27-s + (−4.77 − 1.27i)29-s + (4.50 − 7.79i)31-s + ⋯
L(s)  = 1  + (−0.485 + 0.874i)3-s + (−0.0309 + 0.00828i)5-s + (−0.484 + 0.279i)7-s + (−0.528 − 0.849i)9-s + (−0.177 + 0.660i)11-s + (0.0288 + 0.107i)13-s + (0.00777 − 0.0310i)15-s − 0.206·17-s + (−0.860 + 0.860i)19-s + (−0.00917 − 0.559i)21-s + (−1.55 − 0.896i)23-s + (−0.865 + 0.499i)25-s + (0.998 − 0.0491i)27-s + (−0.886 − 0.237i)29-s + (0.808 − 1.40i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.978 + 0.206i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.978 + 0.206i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0355638 - 0.341389i\)
\(L(\frac12)\) \(\approx\) \(0.0355638 - 0.341389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 - 1.51i)T \)
good5 \( 1 + (0.0691 - 0.0185i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.28 - 0.740i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.587 - 2.19i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.104 - 0.388i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 0.851T + 17T^{2} \)
19 \( 1 + (3.75 - 3.75i)T - 19iT^{2} \)
23 \( 1 + (7.44 + 4.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.77 + 1.27i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.50 + 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.13 - 4.13i)T + 37iT^{2} \)
41 \( 1 + (2.05 + 1.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.669 + 2.49i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.42 + 5.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.95 - 3.95i)T + 53iT^{2} \)
59 \( 1 + (13.7 - 3.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.28 + 0.881i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.73 - 6.47i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.362iT - 71T^{2} \)
73 \( 1 - 15.8iT - 73T^{2} \)
79 \( 1 + (-5.45 - 9.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.37 - 1.43i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 + (0.627 + 1.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.15084078601384864774604413853, −10.05100604245431241046424264325, −9.805803777975758517561821073385, −8.688384975962103873402039696416, −7.71988566276285290583211985230, −6.33444950079547520444509939658, −5.82751321811418804002299907315, −4.50586033823193749278038985474, −3.80324410157332105146294352926, −2.28254119281051074696331494720, 0.19250153808486700776925015464, 1.93131629617346078287082061314, 3.31560553659816801323480711888, 4.70928176449187364941299464866, 5.94378976260316405206430219657, 6.48180517338455658134377088544, 7.58232629313885040648683464216, 8.271778364953279913832513272924, 9.375041437096744652232474214271, 10.46121019481250488813938991390

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