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Properties

Label 2-1152-8.5-c3-0-31
Degree $2$
Conductor $1152$
Sign $0.707 - 0.707i$
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  + 6i·5-s + 21.1·7-s + 42.3i·11-s − 20i·13-s − 8·17-s − 84.6i·19-s + 169.·23-s + 89·25-s + 46i·29-s − 21.1·31-s + 126. i·35-s − 164i·37-s + 312·41-s + 423. i·43-s − 169.·47-s + ⋯
L(s)  = 1  + 0.536i·5-s + 1.14·7-s + 1.16i·11-s − 0.426i·13-s − 0.114·17-s − 1.02i·19-s + 1.53·23-s + 0.711·25-s + 0.294i·29-s − 0.122·31-s + 0.613i·35-s − 0.728i·37-s + 1.18·41-s + 1.50i·43-s − 0.525·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.567375465\)
\(L(\frac12)\) \(\approx\) \(2.567375465\)
\(L(\frac{5}{2})\) 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 \( 1 - 6iT - 125T^{2} \)
7 \( 1 - 21.1T + 343T^{2} \)
11 \( 1 - 42.3iT - 1.33e3T^{2} \)
13 \( 1 + 20iT - 2.19e3T^{2} \)
17 \( 1 + 8T + 4.91e3T^{2} \)
19 \( 1 + 84.6iT - 6.85e3T^{2} \)
23 \( 1 - 169.T + 1.21e4T^{2} \)
29 \( 1 - 46iT - 2.43e4T^{2} \)
31 \( 1 + 21.1T + 2.97e4T^{2} \)
37 \( 1 + 164iT - 5.06e4T^{2} \)
41 \( 1 - 312T + 6.89e4T^{2} \)
43 \( 1 - 423. iT - 7.95e4T^{2} \)
47 \( 1 + 169.T + 1.03e5T^{2} \)
53 \( 1 - 266iT - 1.48e5T^{2} \)
59 \( 1 + 253. iT - 2.05e5T^{2} \)
61 \( 1 - 132iT - 2.26e5T^{2} \)
67 \( 1 + 507. iT - 3.00e5T^{2} \)
71 \( 1 + 677.T + 3.57e5T^{2} \)
73 \( 1 + 246T + 3.89e5T^{2} \)
79 \( 1 - 232.T + 4.93e5T^{2} \)
83 \( 1 + 973. iT - 5.71e5T^{2} \)
89 \( 1 - 1.39e3T + 7.04e5T^{2} \)
97 \( 1 + 302T + 9.12e5T^{2} \)
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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

−9.428835738369168115366466927328, −8.788701521465357771514201068547, −7.65418099098983203744304805846, −7.23686578783479364261199594819, −6.26186390420236299262986898798, −4.94998026026571007814556191358, −4.63306543984598574957539009257, −3.15862577324839819390455785523, −2.20643780495507675980572054619, −1.01512033486156650575088197395, 0.76254989645313024606813594334, 1.68938816589699488749758997620, 3.04778070124184201568645686863, 4.18917185355927322950393102587, 5.05403635175349150163910024257, 5.77377340618477275039220994881, 6.86247919410912032856713318743, 7.84444744088947442729253045814, 8.579720491216612238277776743613, 9.032959481127559728152602678245

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