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Properties

Label 2-1152-48.35-c3-0-42
Degree $2$
Conductor $1152$
Sign $0.925 + 0.378i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.28 + 7.28i)5-s + 29.9·7-s + (0.408 − 0.408i)11-s + (26.6 + 26.6i)13-s − 83.0i·17-s + (51.6 − 51.6i)19-s − 173. i·23-s − 18.7i·25-s + (167. − 167. i)29-s − 191. i·31-s + (218. + 218. i)35-s + (−185. + 185. i)37-s − 62.7·41-s + (193. + 193. i)43-s + 93.1·47-s + ⋯
L(s)  = 1  + (0.651 + 0.651i)5-s + 1.61·7-s + (0.0111 − 0.0111i)11-s + (0.568 + 0.568i)13-s − 1.18i·17-s + (0.623 − 0.623i)19-s − 1.57i·23-s − 0.150i·25-s + (1.07 − 1.07i)29-s − 1.10i·31-s + (1.05 + 1.05i)35-s + (−0.823 + 0.823i)37-s − 0.239·41-s + (0.685 + 0.685i)43-s + 0.288·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.925 + 0.378i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.208556139\)
\(L(\frac12)\) \(\approx\) \(3.208556139\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.28 - 7.28i)T + 125iT^{2} \)
7 \( 1 - 29.9T + 343T^{2} \)
11 \( 1 + (-0.408 + 0.408i)T - 1.33e3iT^{2} \)
13 \( 1 + (-26.6 - 26.6i)T + 2.19e3iT^{2} \)
17 \( 1 + 83.0iT - 4.91e3T^{2} \)
19 \( 1 + (-51.6 + 51.6i)T - 6.85e3iT^{2} \)
23 \( 1 + 173. iT - 1.21e4T^{2} \)
29 \( 1 + (-167. + 167. i)T - 2.43e4iT^{2} \)
31 \( 1 + 191. iT - 2.97e4T^{2} \)
37 \( 1 + (185. - 185. i)T - 5.06e4iT^{2} \)
41 \( 1 + 62.7T + 6.89e4T^{2} \)
43 \( 1 + (-193. - 193. i)T + 7.95e4iT^{2} \)
47 \( 1 - 93.1T + 1.03e5T^{2} \)
53 \( 1 + (249. + 249. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-24.6 + 24.6i)T - 2.05e5iT^{2} \)
61 \( 1 + (451. + 451. i)T + 2.26e5iT^{2} \)
67 \( 1 + (453. - 453. i)T - 3.00e5iT^{2} \)
71 \( 1 + 348. iT - 3.57e5T^{2} \)
73 \( 1 + 923. iT - 3.89e5T^{2} \)
79 \( 1 - 989. iT - 4.93e5T^{2} \)
83 \( 1 + (-325. - 325. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 - 997.T + 9.12e5T^{2} \)
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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

−9.390696323068416169870251263272, −8.491708077389187726897642962844, −7.79128260750734235118748381444, −6.81254853737503883541463529181, −6.09394480228387975281463719053, −4.93797211086662404368485072703, −4.39832970700126478742504202608, −2.83230404240383250075884702720, −2.04909895197309012719853584867, −0.823603605868425626200877175255, 1.38133357034166059104729789853, 1.57074039110600760437881744467, 3.28041945878947543077638741716, 4.38795863914687241680103947584, 5.44102512404658656839628775436, 5.66001618335597605179632439006, 7.12835031004941750968978455361, 7.995680705589902740686698744384, 8.606760530708987525180723300345, 9.307581189760688695096396073542

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