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Properties

Label 2-1152-8.3-c2-0-24
Degree $2$
Conductor $1152$
Sign $0.707 + 0.707i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
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  − 8.89i·5-s + 2.82i·7-s + 18.2·11-s + 5.79i·13-s + 21.5·17-s + 18.2·19-s + 33.3i·23-s − 54.1·25-s − 4.49i·29-s + 2.25i·31-s + 25.1·35-s + 43.1i·37-s + 1.59·41-s + 63.4·43-s − 72.3i·47-s + ⋯
L(s)  = 1  − 1.77i·5-s + 0.404i·7-s + 1.65·11-s + 0.445i·13-s + 1.27·17-s + 0.960·19-s + 1.45i·23-s − 2.16·25-s − 0.154i·29-s + 0.0728i·31-s + 0.719·35-s + 1.16i·37-s + 0.0389·41-s + 1.47·43-s − 1.54i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.336487120\)
\(L(\frac12)\) \(\approx\) \(2.336487120\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + 8.89iT - 25T^{2} \)
7 \( 1 - 2.82iT - 49T^{2} \)
11 \( 1 - 18.2T + 121T^{2} \)
13 \( 1 - 5.79iT - 169T^{2} \)
17 \( 1 - 21.5T + 289T^{2} \)
19 \( 1 - 18.2T + 361T^{2} \)
23 \( 1 - 33.3iT - 529T^{2} \)
29 \( 1 + 4.49iT - 841T^{2} \)
31 \( 1 - 2.25iT - 961T^{2} \)
37 \( 1 - 43.1iT - 1.36e3T^{2} \)
41 \( 1 - 1.59T + 1.68e3T^{2} \)
43 \( 1 - 63.4T + 1.84e3T^{2} \)
47 \( 1 + 72.3iT - 2.20e3T^{2} \)
53 \( 1 + 70.2iT - 2.80e3T^{2} \)
59 \( 1 + 34.6T + 3.48e3T^{2} \)
61 \( 1 - 63.5iT - 3.72e3T^{2} \)
67 \( 1 + 3.24T + 4.48e3T^{2} \)
71 \( 1 + 68.4iT - 5.04e3T^{2} \)
73 \( 1 + 10T + 5.32e3T^{2} \)
79 \( 1 - 35.0iT - 6.24e3T^{2} \)
83 \( 1 + 42.2T + 6.88e3T^{2} \)
89 \( 1 + 5.19T + 7.92e3T^{2} \)
97 \( 1 + 26.8T + 9.40e3T^{2} \)
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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

−9.387499826079486395037844971586, −8.849074562216939269799897209054, −8.027502805514752451268130651630, −7.09295545099273041011637997575, −5.85023452619564536811218642111, −5.30862488780455996313695042156, −4.29712521649249451832474352222, −3.47302918085435419570637535398, −1.63932116951931649320045668130, −0.972867702153550166053926435609, 1.05346163124280722594841766748, 2.59992719157262502180371709321, 3.44028561851027006988067331774, 4.21986487815661520141164261589, 5.78410248829106768565856093801, 6.40071983704680985141695519482, 7.26447083370370928464562267801, 7.72438488216491468128042641080, 9.067466103105459516919195267867, 9.814243143866425657543589641928

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