Properties

Degree $2$
Conductor $1152$
Sign $-0.324 - 0.946i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.32 + 3.32i)5-s − 4.04·7-s + (6.82 − 6.82i)11-s + (−4.29 + 4.29i)13-s − 30.1·17-s + (19.7 + 19.7i)19-s + 28.2·23-s − 2.86i·25-s + (−21.3 + 21.3i)29-s + 38.0i·31-s + (−13.4 − 13.4i)35-s + (42.8 + 42.8i)37-s − 48.2i·41-s + (−32.6 + 32.6i)43-s + 15.8i·47-s + ⋯
L(s)  = 1  + (0.665 + 0.665i)5-s − 0.577·7-s + (0.620 − 0.620i)11-s + (−0.330 + 0.330i)13-s − 1.77·17-s + (1.03 + 1.03i)19-s + 1.22·23-s − 0.114i·25-s + (−0.736 + 0.736i)29-s + 1.22i·31-s + (−0.384 − 0.384i)35-s + (1.15 + 1.15i)37-s − 1.17i·41-s + (−0.759 + 0.759i)43-s + 0.336i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 - 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.324 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.324 - 0.946i$
Motivic weight: \(2\)
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.324 - 0.946i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.483748545\)
\(L(\frac12)\) \(\approx\) \(1.483748545\)
\(L(2)\) 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 \( 1 + (-3.32 - 3.32i)T + 25iT^{2} \)
7 \( 1 + 4.04T + 49T^{2} \)
11 \( 1 + (-6.82 + 6.82i)T - 121iT^{2} \)
13 \( 1 + (4.29 - 4.29i)T - 169iT^{2} \)
17 \( 1 + 30.1T + 289T^{2} \)
19 \( 1 + (-19.7 - 19.7i)T + 361iT^{2} \)
23 \( 1 - 28.2T + 529T^{2} \)
29 \( 1 + (21.3 - 21.3i)T - 841iT^{2} \)
31 \( 1 - 38.0iT - 961T^{2} \)
37 \( 1 + (-42.8 - 42.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (32.6 - 32.6i)T - 1.84e3iT^{2} \)
47 \( 1 - 15.8iT - 2.20e3T^{2} \)
53 \( 1 + (0.476 + 0.476i)T + 2.80e3iT^{2} \)
59 \( 1 + (-9.97 + 9.97i)T - 3.48e3iT^{2} \)
61 \( 1 + (37.9 - 37.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (20.0 + 20.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 40.0T + 5.04e3T^{2} \)
73 \( 1 - 30.8iT - 5.32e3T^{2} \)
79 \( 1 - 130. iT - 6.24e3T^{2} \)
83 \( 1 + (2.26 + 2.26i)T + 6.88e3iT^{2} \)
89 \( 1 - 72.2iT - 7.92e3T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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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

−9.737564620311732457373217209594, −9.200572482950069009079143672454, −8.348881872353473826349628494496, −7.03713489524879649675310678921, −6.62464985009653395625260025684, −5.79566211585722488552790123969, −4.71357108469345924566544221576, −3.49926500422904746976671494515, −2.68364952263024548611170505726, −1.39375296383116913231068436262, 0.44142911537820883525086112630, 1.85284537543137185336629613286, 2.91259413885066386545139979760, 4.27276761688209386486342514352, 5.00932719170165790990987679917, 5.99260854355606202527088716291, 6.84465665263159403409005984506, 7.58459126854219687185336689228, 8.906627843706521713873654588682, 9.338657796654066787212245196170

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