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Properties

Label 2-1152-48.35-c3-0-14
Degree $2$
Conductor $1152$
Sign $-0.567 - 0.823i$
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  + (11.8 + 11.8i)5-s + 12.7·7-s + (15.5 − 15.5i)11-s + (−42.1 − 42.1i)13-s + 76.6i·17-s + (−108. + 108. i)19-s − 52.0i·23-s + 155. i·25-s + (−89.7 + 89.7i)29-s + 12.9i·31-s + (151. + 151. i)35-s + (−115. + 115. i)37-s − 307.·41-s + (342. + 342. i)43-s + 357.·47-s + ⋯
L(s)  = 1  + (1.05 + 1.05i)5-s + 0.688·7-s + (0.425 − 0.425i)11-s + (−0.900 − 0.900i)13-s + 1.09i·17-s + (−1.31 + 1.31i)19-s − 0.471i·23-s + 1.24i·25-s + (−0.574 + 0.574i)29-s + 0.0750i·31-s + (0.729 + 0.729i)35-s + (−0.511 + 0.511i)37-s − 1.17·41-s + (1.21 + 1.21i)43-s + 1.10·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.567 - 0.823i$
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: \((2,\ 1152,\ (\ :3/2),\ -0.567 - 0.823i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.932544162\)
\(L(\frac12)\) \(\approx\) \(1.932544162\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.8 - 11.8i)T + 125iT^{2} \)
7 \( 1 - 12.7T + 343T^{2} \)
11 \( 1 + (-15.5 + 15.5i)T - 1.33e3iT^{2} \)
13 \( 1 + (42.1 + 42.1i)T + 2.19e3iT^{2} \)
17 \( 1 - 76.6iT - 4.91e3T^{2} \)
19 \( 1 + (108. - 108. i)T - 6.85e3iT^{2} \)
23 \( 1 + 52.0iT - 1.21e4T^{2} \)
29 \( 1 + (89.7 - 89.7i)T - 2.43e4iT^{2} \)
31 \( 1 - 12.9iT - 2.97e4T^{2} \)
37 \( 1 + (115. - 115. i)T - 5.06e4iT^{2} \)
41 \( 1 + 307.T + 6.89e4T^{2} \)
43 \( 1 + (-342. - 342. i)T + 7.95e4iT^{2} \)
47 \( 1 - 357.T + 1.03e5T^{2} \)
53 \( 1 + (-450. - 450. i)T + 1.48e5iT^{2} \)
59 \( 1 + (395. - 395. i)T - 2.05e5iT^{2} \)
61 \( 1 + (220. + 220. i)T + 2.26e5iT^{2} \)
67 \( 1 + (243. - 243. i)T - 3.00e5iT^{2} \)
71 \( 1 + 414. iT - 3.57e5T^{2} \)
73 \( 1 + 91.5iT - 3.89e5T^{2} \)
79 \( 1 + 236. iT - 4.93e5T^{2} \)
83 \( 1 + (-204. - 204. i)T + 5.71e5iT^{2} \)
89 \( 1 + 688.T + 7.04e5T^{2} \)
97 \( 1 - 968.T + 9.12e5T^{2} \)
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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

−9.946606442198495753837642495481, −8.853661458033606261939380450528, −8.069895068476220834200817368045, −7.19041373750858939802547135467, −6.14261033432973031287170139320, −5.79410250382251470652201358984, −4.54526197466668924355040844435, −3.39149438839883100348013817521, −2.35060638028448489095122502532, −1.48475638287063541242049479512, 0.41725132688925116934656590338, 1.77367880596729053901748143094, 2.34973959814705514749324889155, 4.19924508142822187319244280403, 4.87311843900294324574861564848, 5.51335616114286485913310350378, 6.70082048278982091267030613996, 7.37878455444555054666968234053, 8.614526166917856440062395522983, 9.190178964541856712051014863091

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