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Properties

Label 2-1152-96.35-c1-0-2
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 $0$

Origins

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Normalization:  

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: $\chi_{1152} (431, \cdot )$
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

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

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