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Properties

Label 2-1152-48.35-c3-0-36
Degree $2$
Conductor $1152$
Sign $0.0398 + 0.999i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.43 + 3.43i)5-s + 8.14·7-s + (−1.92 + 1.92i)11-s + (−9.16 − 9.16i)13-s − 28.6i·17-s + (−39.9 + 39.9i)19-s − 80.3i·23-s − 101. i·25-s + (−113. + 113. i)29-s − 306. i·31-s + (27.9 + 27.9i)35-s + (−47.8 + 47.8i)37-s + 349.·41-s + (164. + 164. i)43-s − 40.5·47-s + ⋯
L(s)  = 1  + (0.307 + 0.307i)5-s + 0.439·7-s + (−0.0526 + 0.0526i)11-s + (−0.195 − 0.195i)13-s − 0.409i·17-s + (−0.481 + 0.481i)19-s − 0.728i·23-s − 0.811i·25-s + (−0.729 + 0.729i)29-s − 1.77i·31-s + (0.135 + 0.135i)35-s + (−0.212 + 0.212i)37-s + 1.33·41-s + (0.583 + 0.583i)43-s − 0.125·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0398 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0398 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.0398 + 0.999i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.0398 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.567222406\)
\(L(\frac12)\) \(\approx\) \(1.567222406\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.43 - 3.43i)T + 125iT^{2} \)
7 \( 1 - 8.14T + 343T^{2} \)
11 \( 1 + (1.92 - 1.92i)T - 1.33e3iT^{2} \)
13 \( 1 + (9.16 + 9.16i)T + 2.19e3iT^{2} \)
17 \( 1 + 28.6iT - 4.91e3T^{2} \)
19 \( 1 + (39.9 - 39.9i)T - 6.85e3iT^{2} \)
23 \( 1 + 80.3iT - 1.21e4T^{2} \)
29 \( 1 + (113. - 113. i)T - 2.43e4iT^{2} \)
31 \( 1 + 306. iT - 2.97e4T^{2} \)
37 \( 1 + (47.8 - 47.8i)T - 5.06e4iT^{2} \)
41 \( 1 - 349.T + 6.89e4T^{2} \)
43 \( 1 + (-164. - 164. i)T + 7.95e4iT^{2} \)
47 \( 1 + 40.5T + 1.03e5T^{2} \)
53 \( 1 + (454. + 454. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-245. + 245. i)T - 2.05e5iT^{2} \)
61 \( 1 + (186. + 186. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-118. + 118. i)T - 3.00e5iT^{2} \)
71 \( 1 - 414. iT - 3.57e5T^{2} \)
73 \( 1 - 431. iT - 3.89e5T^{2} \)
79 \( 1 + 1.04e3iT - 4.93e5T^{2} \)
83 \( 1 + (-739. - 739. i)T + 5.71e5iT^{2} \)
89 \( 1 - 942.T + 7.04e5T^{2} \)
97 \( 1 + 983.T + 9.12e5T^{2} \)
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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

−9.352281252359535956534666883374, −8.274383094981120451536707433028, −7.69034329528909942366618203779, −6.64365717767624067351083482266, −5.90697316495759756497298376899, −4.90563289163918298597900725434, −4.01465617137001372239481182792, −2.76474136642956514159294407438, −1.85368118605926740214978280204, −0.38024920159275566796045094763, 1.18755637799539841669138070937, 2.18363252808702928273606699336, 3.45284574824493818566890558834, 4.52627669601246683608049031051, 5.34283051366668508994449142468, 6.19303504508960668427493591358, 7.22999716313307107918360832036, 7.952636783798699938218788062466, 8.967956990655250679831843028009, 9.412294086994091453715402419042

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