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Properties

Label 2-1152-48.5-c2-0-8
Degree $2$
Conductor $1152$
Sign $-0.653 + 0.757i$
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  + (−6.08 + 6.08i)5-s + 9.40i·7-s + (−11.9 + 11.9i)11-s + (−3.47 + 3.47i)13-s + 28.5i·17-s + (−3.08 + 3.08i)19-s + 2.57·23-s − 49.1i·25-s + (3.49 + 3.49i)29-s + 21.0·31-s + (−57.2 − 57.2i)35-s + (13.2 + 13.2i)37-s + 11.2·41-s + (8.19 + 8.19i)43-s − 17.2i·47-s + ⋯
L(s)  = 1  + (−1.21 + 1.21i)5-s + 1.34i·7-s + (−1.08 + 1.08i)11-s + (−0.267 + 0.267i)13-s + 1.67i·17-s + (−0.162 + 0.162i)19-s + 0.111·23-s − 1.96i·25-s + (0.120 + 0.120i)29-s + 0.680·31-s + (−1.63 − 1.63i)35-s + (0.359 + 0.359i)37-s + 0.274·41-s + (0.190 + 0.190i)43-s − 0.366i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.653 + 0.757i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.653 + 0.757i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8224549557\)
\(L(\frac12)\) \(\approx\) \(0.8224549557\)
\(L(2)\) 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 \( 1 + (6.08 - 6.08i)T - 25iT^{2} \)
7 \( 1 - 9.40iT - 49T^{2} \)
11 \( 1 + (11.9 - 11.9i)T - 121iT^{2} \)
13 \( 1 + (3.47 - 3.47i)T - 169iT^{2} \)
17 \( 1 - 28.5iT - 289T^{2} \)
19 \( 1 + (3.08 - 3.08i)T - 361iT^{2} \)
23 \( 1 - 2.57T + 529T^{2} \)
29 \( 1 + (-3.49 - 3.49i)T + 841iT^{2} \)
31 \( 1 - 21.0T + 961T^{2} \)
37 \( 1 + (-13.2 - 13.2i)T + 1.36e3iT^{2} \)
41 \( 1 - 11.2T + 1.68e3T^{2} \)
43 \( 1 + (-8.19 - 8.19i)T + 1.84e3iT^{2} \)
47 \( 1 + 17.2iT - 2.20e3T^{2} \)
53 \( 1 + (30.5 - 30.5i)T - 2.80e3iT^{2} \)
59 \( 1 + (14.7 - 14.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (39.3 - 39.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (-79.6 + 79.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 73.1T + 5.04e3T^{2} \)
73 \( 1 - 1.23iT - 5.32e3T^{2} \)
79 \( 1 - 110.T + 6.24e3T^{2} \)
83 \( 1 + (-13.7 - 13.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 39.2T + 7.92e3T^{2} \)
97 \( 1 + 109.T + 9.40e3T^{2} \)
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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

−10.32707395186308345662607758282, −9.322932994761799568245388545260, −8.169760195840436178585427518387, −7.85186862729243200946066932124, −6.81548589461576669999914149420, −6.07609462547951228906779986657, −4.93494365108701094108158866044, −3.95702154615817127308466361215, −2.87521637087761554446119530041, −2.11673575130546710334338701328, 0.34908072576052922705815568005, 0.808446626470454696986857202663, 2.87948425354916534544124052417, 3.87534494982612825745179639262, 4.73692170571232160080278423052, 5.34021235144800585866355994043, 6.79614696979359126156177530816, 7.76795141969762468119248037415, 7.982488264549949654042060398405, 8.989329580739738964201655709781

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