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Properties

Label 2-1152-48.5-c2-0-16
Degree $2$
Conductor $1152$
Sign $0.993 - 0.110i$
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  + (1.84 − 1.84i)5-s − 0.226i·7-s + (−11.5 + 11.5i)11-s + (8.11 − 8.11i)13-s − 6.20i·17-s + (−4.43 + 4.43i)19-s + 28.6·23-s + 18.1i·25-s + (15.7 + 15.7i)29-s + 33.5·31-s + (−0.418 − 0.418i)35-s + (−43.8 − 43.8i)37-s + 62.2·41-s + (18.3 + 18.3i)43-s + 13.8i·47-s + ⋯
L(s)  = 1  + (0.369 − 0.369i)5-s − 0.0323i·7-s + (−1.05 + 1.05i)11-s + (0.624 − 0.624i)13-s − 0.364i·17-s + (−0.233 + 0.233i)19-s + 1.24·23-s + 0.727i·25-s + (0.543 + 0.543i)29-s + 1.08·31-s + (−0.0119 − 0.0119i)35-s + (−1.18 − 1.18i)37-s + 1.51·41-s + (0.425 + 0.425i)43-s + 0.293i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.993 - 0.110i$
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: \((2,\ 1152,\ (\ :1),\ 0.993 - 0.110i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.003284627\)
\(L(\frac12)\) \(\approx\) \(2.003284627\)
\(L(2)\) 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 \( 1 + (-1.84 + 1.84i)T - 25iT^{2} \)
7 \( 1 + 0.226iT - 49T^{2} \)
11 \( 1 + (11.5 - 11.5i)T - 121iT^{2} \)
13 \( 1 + (-8.11 + 8.11i)T - 169iT^{2} \)
17 \( 1 + 6.20iT - 289T^{2} \)
19 \( 1 + (4.43 - 4.43i)T - 361iT^{2} \)
23 \( 1 - 28.6T + 529T^{2} \)
29 \( 1 + (-15.7 - 15.7i)T + 841iT^{2} \)
31 \( 1 - 33.5T + 961T^{2} \)
37 \( 1 + (43.8 + 43.8i)T + 1.36e3iT^{2} \)
41 \( 1 - 62.2T + 1.68e3T^{2} \)
43 \( 1 + (-18.3 - 18.3i)T + 1.84e3iT^{2} \)
47 \( 1 - 13.8iT - 2.20e3T^{2} \)
53 \( 1 + (37.9 - 37.9i)T - 2.80e3iT^{2} \)
59 \( 1 + (70.4 - 70.4i)T - 3.48e3iT^{2} \)
61 \( 1 + (-60.3 + 60.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (-61.9 + 61.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 32.5T + 5.04e3T^{2} \)
73 \( 1 + 130. iT - 5.32e3T^{2} \)
79 \( 1 - 132.T + 6.24e3T^{2} \)
83 \( 1 + (-19.2 - 19.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 91.2T + 7.92e3T^{2} \)
97 \( 1 + 20.0T + 9.40e3T^{2} \)
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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

−9.509819296358683263408893920630, −8.946671353035630464097859534157, −7.88274483742388893214376386238, −7.28184114707734470690679628923, −6.18757738524011309922635136596, −5.25197378731866166697309458489, −4.63506951009768935713685573526, −3.28792045357657014991126635922, −2.24636432434113010971550580397, −0.924633158868434402124620111738, 0.809061987527933787359199989376, 2.35055200007140327171226402595, 3.19585312025199757051421318536, 4.39403789420532913389700285025, 5.43179325179575882032313229259, 6.26131175327761057100239911274, 6.94712942386818857961483006804, 8.184807074619093045846863462754, 8.600870163732519963067313426992, 9.646428973991087090641962388169

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