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Properties

Label 2-1152-96.35-c1-0-11
Degree $2$
Conductor $1152$
Sign $0.985 + 0.168i$
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  + (1.78 − 0.741i)5-s + (2.33 + 2.33i)7-s + (0.683 − 0.283i)11-s + (2.43 − 5.88i)13-s + 0.967·17-s + (2.52 + 1.04i)19-s + (−5.00 − 5.00i)23-s + (−0.882 + 0.882i)25-s + (−0.563 + 1.36i)29-s + 7.28i·31-s + (5.91 + 2.45i)35-s + (3.26 + 7.88i)37-s + (6.80 − 6.80i)41-s + (−1.36 − 3.29i)43-s − 3.69i·47-s + ⋯
L(s)  = 1  + (0.800 − 0.331i)5-s + (0.883 + 0.883i)7-s + (0.206 − 0.0853i)11-s + (0.675 − 1.63i)13-s + 0.234·17-s + (0.579 + 0.240i)19-s + (−1.04 − 1.04i)23-s + (−0.176 + 0.176i)25-s + (−0.104 + 0.252i)29-s + 1.30i·31-s + (0.999 + 0.414i)35-s + (0.536 + 1.29i)37-s + (1.06 − 1.06i)41-s + (−0.208 − 0.502i)43-s − 0.538i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.985 + 0.168i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.985 + 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.179169560\)
\(L(\frac12)\) \(\approx\) \(2.179169560\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 0.741i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.33 - 2.33i)T + 7iT^{2} \)
11 \( 1 + (-0.683 + 0.283i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-2.43 + 5.88i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.967T + 17T^{2} \)
19 \( 1 + (-2.52 - 1.04i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.00 + 5.00i)T + 23iT^{2} \)
29 \( 1 + (0.563 - 1.36i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 7.28iT - 31T^{2} \)
37 \( 1 + (-3.26 - 7.88i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.80 + 6.80i)T - 41iT^{2} \)
43 \( 1 + (1.36 + 3.29i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.69iT - 47T^{2} \)
53 \( 1 + (-1.85 - 4.47i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.05 + 9.79i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-10.8 - 4.48i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (1.49 - 3.60i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (9.85 - 9.85i)T - 71iT^{2} \)
73 \( 1 + (4.81 + 4.81i)T + 73iT^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + (-1.15 + 2.77i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.82 + 6.82i)T + 89iT^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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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

−9.878995410416653423004583923590, −8.702387073518017024879808192422, −8.416904106143893335323831005124, −7.43447716358995958246402486639, −6.07548393497915397119463713523, −5.60421861178070500246228423789, −4.85427594159109210949323523193, −3.46479192376760902646421010864, −2.32320780916800000983022992029, −1.19360479303644451438080475537, 1.34033057235101097373225930000, 2.24958431237583500517305258306, 3.85782923406133150351827500829, 4.45655251765191012820921063682, 5.74098641607002746807889303283, 6.39857131000507197397405609705, 7.42252603220236445488772234253, 8.004953883668492528440526257648, 9.311126105390064513002615974895, 9.639341006191402399852110071881

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