↑

Properties

Label 2-1152-48.5-c2-0-18
Degree $2$
Conductor $1152$
Sign $0.810 + 0.585i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.14 + 5.14i)5-s − 7.48i·7-s + (−11.6 + 11.6i)11-s + (14.0 − 14.0i)13-s + 7.92i·17-s + (−10.7 + 10.7i)19-s + 2.09·23-s − 27.9i·25-s + (−23.6 − 23.6i)29-s − 17.8·31-s + (38.5 + 38.5i)35-s + (30.8 + 30.8i)37-s + 36.8·41-s + (−28.4 − 28.4i)43-s − 65.3i·47-s + ⋯
L(s)  = 1  + (−1.02 + 1.02i)5-s − 1.06i·7-s + (−1.06 + 1.06i)11-s + (1.08 − 1.08i)13-s + 0.466i·17-s + (−0.567 + 0.567i)19-s + 0.0909·23-s − 1.11i·25-s + (−0.814 − 0.814i)29-s − 0.576·31-s + (1.10 + 1.10i)35-s + (0.834 + 0.834i)37-s + 0.898·41-s + (−0.660 − 0.660i)43-s − 1.39i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.810 + 0.585i$
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: \((2,\ 1152,\ (\ :1),\ 0.810 + 0.585i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.123756921\)
\(L(\frac12)\) \(\approx\) \(1.123756921\)
\(L(2)\) 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 \( 1 + (5.14 - 5.14i)T - 25iT^{2} \)
7 \( 1 + 7.48iT - 49T^{2} \)
11 \( 1 + (11.6 - 11.6i)T - 121iT^{2} \)
13 \( 1 + (-14.0 + 14.0i)T - 169iT^{2} \)
17 \( 1 - 7.92iT - 289T^{2} \)
19 \( 1 + (10.7 - 10.7i)T - 361iT^{2} \)
23 \( 1 - 2.09T + 529T^{2} \)
29 \( 1 + (23.6 + 23.6i)T + 841iT^{2} \)
31 \( 1 + 17.8T + 961T^{2} \)
37 \( 1 + (-30.8 - 30.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 36.8T + 1.68e3T^{2} \)
43 \( 1 + (28.4 + 28.4i)T + 1.84e3iT^{2} \)
47 \( 1 + 65.3iT - 2.20e3T^{2} \)
53 \( 1 + (-9.05 + 9.05i)T - 2.80e3iT^{2} \)
59 \( 1 + (-74.0 + 74.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (-53.4 + 53.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (-20.6 + 20.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 39.6T + 5.04e3T^{2} \)
73 \( 1 - 91.3iT - 5.32e3T^{2} \)
79 \( 1 - 92.1T + 6.24e3T^{2} \)
83 \( 1 + (9.12 + 9.12i)T + 6.88e3iT^{2} \)
89 \( 1 - 63.2T + 7.92e3T^{2} \)
97 \( 1 - 152.T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.891582980918546041225550506992, −8.314722143752423698973611100743, −7.82993161996092065352651522171, −7.19979925228197871452433625643, −6.35693712623088421345390938392, −5.20025815945869801985379687926, −3.93915207091664168711995863159, −3.55306780490279540835123675138, −2.21513329328530052151529499228, −0.46969726025015606938103195493, 0.840141476011420719903914912358, 2.37433399043132519942875058556, 3.56539716810561296603446628193, 4.51405557942554261303398827949, 5.40983105280711873274797515686, 6.15219678252827016172953048531, 7.40989583439478325215518224521, 8.235228764680489128337764474085, 8.902638263535387371664959363003, 9.200238743621199447755756889893

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