Properties

Degree $2$
Conductor $1152$
Sign $0.162 - 0.986i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.40 + 3.40i)5-s + 12.1·7-s + (9.81 + 9.81i)11-s + (7.76 + 7.76i)13-s − 9.73·17-s + (−11.2 + 11.2i)19-s + 20.2·23-s + 1.80i·25-s + (−16.4 − 16.4i)29-s − 26.3i·31-s + (−41.3 + 41.3i)35-s + (23.7 − 23.7i)37-s + 24.7i·41-s + (−29.8 − 29.8i)43-s + 31.3i·47-s + ⋯
L(s)  = 1  + (−0.681 + 0.681i)5-s + 1.73·7-s + (0.891 + 0.891i)11-s + (0.597 + 0.597i)13-s − 0.572·17-s + (−0.593 + 0.593i)19-s + 0.881·23-s + 0.0720i·25-s + (−0.565 − 0.565i)29-s − 0.850i·31-s + (−1.18 + 1.18i)35-s + (0.641 − 0.641i)37-s + 0.603i·41-s + (−0.694 − 0.694i)43-s + 0.666i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.162 - 0.986i$
Motivic weight: \(2\)
Character: $\chi_{1152} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.162 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.119977333\)
\(L(\frac12)\) \(\approx\) \(2.119977333\)
\(L(2)\) 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 \( 1 + (3.40 - 3.40i)T - 25iT^{2} \)
7 \( 1 - 12.1T + 49T^{2} \)
11 \( 1 + (-9.81 - 9.81i)T + 121iT^{2} \)
13 \( 1 + (-7.76 - 7.76i)T + 169iT^{2} \)
17 \( 1 + 9.73T + 289T^{2} \)
19 \( 1 + (11.2 - 11.2i)T - 361iT^{2} \)
23 \( 1 - 20.2T + 529T^{2} \)
29 \( 1 + (16.4 + 16.4i)T + 841iT^{2} \)
31 \( 1 + 26.3iT - 961T^{2} \)
37 \( 1 + (-23.7 + 23.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 24.7iT - 1.68e3T^{2} \)
43 \( 1 + (29.8 + 29.8i)T + 1.84e3iT^{2} \)
47 \( 1 - 31.3iT - 2.20e3T^{2} \)
53 \( 1 + (-36.8 + 36.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (14.1 + 14.1i)T + 3.48e3iT^{2} \)
61 \( 1 + (-42.5 - 42.5i)T + 3.72e3iT^{2} \)
67 \( 1 + (48.7 - 48.7i)T - 4.48e3iT^{2} \)
71 \( 1 + 7.73T + 5.04e3T^{2} \)
73 \( 1 - 85.4iT - 5.32e3T^{2} \)
79 \( 1 - 105. iT - 6.24e3T^{2} \)
83 \( 1 + (62.1 - 62.1i)T - 6.88e3iT^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + 147.T + 9.40e3T^{2} \)
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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

−9.777204694056981309731064168151, −8.831708163241705926897993917222, −8.118143617670273669205584108493, −7.31662892899066934682522686928, −6.67051741067300766259366417079, −5.48384938122749346712492721469, −4.30955238530971907975645890062, −3.97391680134823151783984178821, −2.31030231780519673347782553820, −1.37001621383796925425740362850, 0.71218574991580774252242417584, 1.67181810689547987711191994863, 3.26417529378004335154210391467, 4.37909632225018740007919247041, 4.91000351099602324239181641430, 5.95089941606309338336169980439, 7.05409262528954011117479268509, 8.004752325759354795703738595049, 8.690062860830322480921453731626, 8.907272330475513308110039899408

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