# Properties

 Label 2-1152-96.11-c1-0-15 Degree $2$ Conductor $1152$ Sign $-0.507 + 0.861i$ Analytic cond. $9.19876$ Root an. cond. $3.03294$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.0913 + 0.0378i)5-s + (3.05 − 3.05i)7-s + (−5.25 − 2.17i)11-s + (−1.57 − 3.79i)13-s + 2.56·17-s + (−2.64 + 1.09i)19-s + (−4.03 + 4.03i)23-s + (−3.52 − 3.52i)25-s + (−2.06 − 4.99i)29-s + 1.44i·31-s + (0.394 − 0.163i)35-s + (−2.07 + 5.01i)37-s + (−0.296 − 0.296i)41-s + (2.72 − 6.57i)43-s + 7.42i·47-s + ⋯
 L(s)  = 1 + (0.0408 + 0.0169i)5-s + (1.15 − 1.15i)7-s + (−1.58 − 0.656i)11-s + (−0.436 − 1.05i)13-s + 0.622·17-s + (−0.605 + 0.250i)19-s + (−0.840 + 0.840i)23-s + (−0.705 − 0.705i)25-s + (−0.384 − 0.927i)29-s + 0.258i·31-s + (0.0666 − 0.0276i)35-s + (−0.341 + 0.824i)37-s + (−0.0463 − 0.0463i)41-s + (0.415 − 1.00i)43-s + 1.08i·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1152$$    =    $$2^{7} \cdot 3^{2}$$ Sign: $-0.507 + 0.861i$ Analytic conductor: $$9.19876$$ Root analytic conductor: $$3.03294$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1152} (719, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1152,\ (\ :1/2),\ -0.507 + 0.861i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.165055438$$ $$L(\frac12)$$ $$\approx$$ $$1.165055438$$ $$L(\frac{3}{2})$$ 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 + (-0.0913 - 0.0378i)T + (3.53 + 3.53i)T^{2}$$
7 $$1 + (-3.05 + 3.05i)T - 7iT^{2}$$
11 $$1 + (5.25 + 2.17i)T + (7.77 + 7.77i)T^{2}$$
13 $$1 + (1.57 + 3.79i)T + (-9.19 + 9.19i)T^{2}$$
17 $$1 - 2.56T + 17T^{2}$$
19 $$1 + (2.64 - 1.09i)T + (13.4 - 13.4i)T^{2}$$
23 $$1 + (4.03 - 4.03i)T - 23iT^{2}$$
29 $$1 + (2.06 + 4.99i)T + (-20.5 + 20.5i)T^{2}$$
31 $$1 - 1.44iT - 31T^{2}$$
37 $$1 + (2.07 - 5.01i)T + (-26.1 - 26.1i)T^{2}$$
41 $$1 + (0.296 + 0.296i)T + 41iT^{2}$$
43 $$1 + (-2.72 + 6.57i)T + (-30.4 - 30.4i)T^{2}$$
47 $$1 - 7.42iT - 47T^{2}$$
53 $$1 + (1.53 - 3.69i)T + (-37.4 - 37.4i)T^{2}$$
59 $$1 + (-0.988 + 2.38i)T + (-41.7 - 41.7i)T^{2}$$
61 $$1 + (-10.4 + 4.32i)T + (43.1 - 43.1i)T^{2}$$
67 $$1 + (-0.690 - 1.66i)T + (-47.3 + 47.3i)T^{2}$$
71 $$1 + (2.97 + 2.97i)T + 71iT^{2}$$
73 $$1 + (-9.22 + 9.22i)T - 73iT^{2}$$
79 $$1 - 1.12T + 79T^{2}$$
83 $$1 + (4.13 + 9.98i)T + (-58.6 + 58.6i)T^{2}$$
89 $$1 + (-12.0 + 12.0i)T - 89iT^{2}$$
97 $$1 + 18.5T + 97T^{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

−9.836479282953239759487227167606, −8.266677277749597264593467805441, −7.948562001151850142218803312118, −7.38910908466559507552230242630, −5.94879061833320986066640580652, −5.26621801863298361022483957406, −4.34056491250565302659680143648, −3.26716634128135356293974153070, −2.00635130906972477764343124331, −0.47465296117545333933074714137, 1.93873807550196192369070455103, 2.48104568382684979384159272269, 4.12406821565227681058266306106, 5.12803562322084277474379482278, 5.53856933773335714157383266095, 6.83091140923840060280454332024, 7.77727364255971946031798297844, 8.339951817868413138530966784152, 9.220696881264562036586152109683, 10.04036340445550335480577511242