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Properties

Label 2-1152-16.11-c2-0-26
Degree $2$
Conductor $1152$
Sign $-0.282 + 0.959i$
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  + (0.909 + 0.909i)5-s − 0.654·7-s + (−13.3 + 13.3i)11-s + (−8.32 + 8.32i)13-s + 3.93·17-s + (−16.8 − 16.8i)19-s + 23.1·23-s − 23.3i·25-s + (35.6 − 35.6i)29-s − 45.5i·31-s + (−0.595 − 0.595i)35-s + (−10.1 − 10.1i)37-s − 28.4i·41-s + (−22.7 + 22.7i)43-s + 10.7i·47-s + ⋯
L(s)  = 1  + (0.181 + 0.181i)5-s − 0.0935·7-s + (−1.21 + 1.21i)11-s + (−0.640 + 0.640i)13-s + 0.231·17-s + (−0.889 − 0.889i)19-s + 1.00·23-s − 0.933i·25-s + (1.22 − 1.22i)29-s − 1.46i·31-s + (−0.0170 − 0.0170i)35-s + (−0.274 − 0.274i)37-s − 0.694i·41-s + (−0.528 + 0.528i)43-s + 0.229i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.282 + 0.959i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.282 + 0.959i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7835089554\)
\(L(\frac12)\) \(\approx\) \(0.7835089554\)
\(L(2)\) 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 \( 1 + (-0.909 - 0.909i)T + 25iT^{2} \)
7 \( 1 + 0.654T + 49T^{2} \)
11 \( 1 + (13.3 - 13.3i)T - 121iT^{2} \)
13 \( 1 + (8.32 - 8.32i)T - 169iT^{2} \)
17 \( 1 - 3.93T + 289T^{2} \)
19 \( 1 + (16.8 + 16.8i)T + 361iT^{2} \)
23 \( 1 - 23.1T + 529T^{2} \)
29 \( 1 + (-35.6 + 35.6i)T - 841iT^{2} \)
31 \( 1 + 45.5iT - 961T^{2} \)
37 \( 1 + (10.1 + 10.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 28.4iT - 1.68e3T^{2} \)
43 \( 1 + (22.7 - 22.7i)T - 1.84e3iT^{2} \)
47 \( 1 - 10.7iT - 2.20e3T^{2} \)
53 \( 1 + (-41.5 - 41.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (21.0 - 21.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (-68.7 + 68.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (67.8 + 67.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 33.3T + 5.04e3T^{2} \)
73 \( 1 - 18.6iT - 5.32e3T^{2} \)
79 \( 1 + 6.29iT - 6.24e3T^{2} \)
83 \( 1 + (72.0 + 72.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 10.6iT - 7.92e3T^{2} \)
97 \( 1 - 143.T + 9.40e3T^{2} \)
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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

−9.556738162260519488578715937303, −8.522578173092034215759783528028, −7.63053513233868545769990733625, −6.93147843566354313973680469395, −6.07019629530865203059172639102, −4.84788414181751930303025754746, −4.38850911220633956940604414256, −2.74515979821127332617404631542, −2.12847497979906925455151600589, −0.24001345020686397180331951200, 1.21564229173920134007611516096, 2.76003117025891406454459785728, 3.44024033569862817192874861440, 5.00799202097937375209437995137, 5.40224852933918195794823178583, 6.49624134276436998042146026949, 7.41641209827859297280525804073, 8.401193558583083139856516608036, 8.764192060578990068699580617695, 10.13096554029708412346201503151

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