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Properties

Label 2-1152-16.11-c2-0-14
Degree $2$
Conductor $1152$
Sign $-0.336 - 0.941i$
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  + (6.01 + 6.01i)5-s + 8.23·7-s + (−6.51 + 6.51i)11-s + (−8.82 + 8.82i)13-s − 14.1·17-s + (23.7 + 23.7i)19-s − 9.42·23-s + 47.3i·25-s + (−23.7 + 23.7i)29-s − 24.4i·31-s + (49.4 + 49.4i)35-s + (−24.2 − 24.2i)37-s − 6.67i·41-s + (−0.897 + 0.897i)43-s + 25.2i·47-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)5-s + 1.17·7-s + (−0.592 + 0.592i)11-s + (−0.679 + 0.679i)13-s − 0.833·17-s + (1.24 + 1.24i)19-s − 0.409·23-s + 1.89i·25-s + (−0.820 + 0.820i)29-s − 0.787i·31-s + (1.41 + 1.41i)35-s + (−0.654 − 0.654i)37-s − 0.162i·41-s + (−0.0208 + 0.0208i)43-s + 0.537i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.336 - 0.941i$
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: \((2,\ 1152,\ (\ :1),\ -0.336 - 0.941i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.316525415\)
\(L(\frac12)\) \(\approx\) \(2.316525415\)
\(L(2)\) 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 \( 1 + (-6.01 - 6.01i)T + 25iT^{2} \)
7 \( 1 - 8.23T + 49T^{2} \)
11 \( 1 + (6.51 - 6.51i)T - 121iT^{2} \)
13 \( 1 + (8.82 - 8.82i)T - 169iT^{2} \)
17 \( 1 + 14.1T + 289T^{2} \)
19 \( 1 + (-23.7 - 23.7i)T + 361iT^{2} \)
23 \( 1 + 9.42T + 529T^{2} \)
29 \( 1 + (23.7 - 23.7i)T - 841iT^{2} \)
31 \( 1 + 24.4iT - 961T^{2} \)
37 \( 1 + (24.2 + 24.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 6.67iT - 1.68e3T^{2} \)
43 \( 1 + (0.897 - 0.897i)T - 1.84e3iT^{2} \)
47 \( 1 - 25.2iT - 2.20e3T^{2} \)
53 \( 1 + (32.6 + 32.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (-8.31 + 8.31i)T - 3.48e3iT^{2} \)
61 \( 1 + (-68.4 + 68.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (-7.71 - 7.71i)T + 4.48e3iT^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + 52.8iT - 5.32e3T^{2} \)
79 \( 1 + 87.2iT - 6.24e3T^{2} \)
83 \( 1 + (9.53 + 9.53i)T + 6.88e3iT^{2} \)
89 \( 1 - 146. iT - 7.92e3T^{2} \)
97 \( 1 - 101.T + 9.40e3T^{2} \)
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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

−9.812734817042655510138119350933, −9.345455812331498800619171246771, −8.020006236393686962215311440468, −7.35526967386383851403017661210, −6.57746148410661749138404163411, −5.56190574060914836153490145580, −4.91230327875446686469134300003, −3.59552078370010260077189838047, −2.23404859287974762619046055331, −1.81825993321979048951539843125, 0.65082064522622516917759229827, 1.77613828901232068054333899684, 2.76300737202315579639436965950, 4.45252488942499922808531115546, 5.28522931054147573981229971609, 5.49558530131556989542149891699, 6.86444188455994062089302153422, 7.952467555594906432701367979994, 8.541133410575707894100994942282, 9.339143623931115186546241382746

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