Properties

Label 2-24e2-8.5-c1-0-1
Degree $2$
Conductor $576$
Sign $-0.258 - 0.965i$
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  − 3.46·7-s + 6.92i·13-s + 8i·19-s + 5·25-s − 10.3·31-s + 6.92i·37-s − 8i·43-s + 4.99·49-s − 6.92i·61-s + 16i·67-s − 10·73-s + 17.3·79-s − 23.9i·91-s − 14·97-s + 3.46·103-s + ⋯
L(s)  = 1  − 1.30·7-s + 1.92i·13-s + 1.83i·19-s + 25-s − 1.86·31-s + 1.13i·37-s − 1.21i·43-s + 0.714·49-s − 0.887i·61-s + 1.95i·67-s − 1.17·73-s + 1.94·79-s − 2.51i·91-s − 1.42·97-s + 0.341·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531857 + 0.693130i\)
\(L(\frac12)\) \(\approx\) \(0.531857 + 0.693130i\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 16iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.92360165800917855017185872965, −9.965066290601168683592973602306, −9.328342953022103823308423458819, −8.501974515438193711159163136782, −7.17200878518640598048076687065, −6.56056507341188421033187019462, −5.60454693503504828223059983383, −4.19970158031309724699419925657, −3.34173539391058334856183077115, −1.81762917176668932327044231594, 0.48177600340166529814338386913, 2.71705476216629341101441829728, 3.46881006421485080198859346656, 4.96571226389503390515010892979, 5.89352805678870864365951215710, 6.87858262977726324973536900542, 7.69312328104009617759245389385, 8.905868698652704995489566248196, 9.519175689722656529341327941221, 10.55487254875962330697560427659

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