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Properties

Label 2-1152-48.5-c2-0-5
Degree $2$
Conductor $1152$
Sign $-0.404 - 0.914i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.900 + 0.900i)5-s − 7.66i·7-s + (4.57 − 4.57i)11-s + (−10.9 + 10.9i)13-s − 0.0435i·17-s + (−12.9 + 12.9i)19-s − 18.3·23-s + 23.3i·25-s + (−34.9 − 34.9i)29-s + 38.4·31-s + (6.89 + 6.89i)35-s + (−11.3 − 11.3i)37-s + 45.6·41-s + (51.4 + 51.4i)43-s + 56.5i·47-s + ⋯
L(s)  = 1  + (−0.180 + 0.180i)5-s − 1.09i·7-s + (0.415 − 0.415i)11-s + (−0.844 + 0.844i)13-s − 0.00256i·17-s + (−0.683 + 0.683i)19-s − 0.796·23-s + 0.935i·25-s + (−1.20 − 1.20i)29-s + 1.24·31-s + (0.197 + 0.197i)35-s + (−0.307 − 0.307i)37-s + 1.11·41-s + (1.19 + 1.19i)43-s + 1.20i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.404 - 0.914i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.404 - 0.914i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7528625080\)
\(L(\frac12)\) \(\approx\) \(0.7528625080\)
\(L(2)\) 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 \( 1 + (0.900 - 0.900i)T - 25iT^{2} \)
7 \( 1 + 7.66iT - 49T^{2} \)
11 \( 1 + (-4.57 + 4.57i)T - 121iT^{2} \)
13 \( 1 + (10.9 - 10.9i)T - 169iT^{2} \)
17 \( 1 + 0.0435iT - 289T^{2} \)
19 \( 1 + (12.9 - 12.9i)T - 361iT^{2} \)
23 \( 1 + 18.3T + 529T^{2} \)
29 \( 1 + (34.9 + 34.9i)T + 841iT^{2} \)
31 \( 1 - 38.4T + 961T^{2} \)
37 \( 1 + (11.3 + 11.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 45.6T + 1.68e3T^{2} \)
43 \( 1 + (-51.4 - 51.4i)T + 1.84e3iT^{2} \)
47 \( 1 - 56.5iT - 2.20e3T^{2} \)
53 \( 1 + (44.6 - 44.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (20.7 - 20.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (-2.42 + 2.42i)T - 3.72e3iT^{2} \)
67 \( 1 + (-18.9 + 18.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 114.T + 5.04e3T^{2} \)
73 \( 1 - 127. iT - 5.32e3T^{2} \)
79 \( 1 - 37.0T + 6.24e3T^{2} \)
83 \( 1 + (-84.4 - 84.4i)T + 6.88e3iT^{2} \)
89 \( 1 + 136.T + 7.92e3T^{2} \)
97 \( 1 + 173.T + 9.40e3T^{2} \)
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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

−9.765552868267570184137814395880, −9.256875932683279666817882663827, −7.977081533528615669426874726399, −7.50446117125038554311962509051, −6.55963441687535120072469477572, −5.78322712516162483787598976448, −4.31502911262971628624777017084, −4.01012511934009662660603072960, −2.60285145276033962770277603755, −1.26921254925727537731104805541, 0.22984589963073946204548957740, 1.98869866042801566621208640713, 2.86638720744394784236857915862, 4.17252647149222385049053962901, 5.08937118574067516578337694299, 5.89164233907354396228452612786, 6.84441530520266971758050637620, 7.77853668885087481373761504223, 8.603579598417664447353443363259, 9.256521135179004275220823186631

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