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Properties

Label 2-1152-16.11-c2-0-27
Degree $2$
Conductor $1152$
Sign $-0.548 + 0.835i$
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  + (−0.0586 − 0.0586i)5-s − 4.61·7-s + (5.36 − 5.36i)11-s + (−11.0 + 11.0i)13-s + 12.8·17-s + (2.63 + 2.63i)19-s + 16.3·23-s − 24.9i·25-s + (−26.0 + 26.0i)29-s − 20.2i·31-s + (0.270 + 0.270i)35-s + (−41.2 − 41.2i)37-s − 3.29i·41-s + (−0.786 + 0.786i)43-s − 79.7i·47-s + ⋯
L(s)  = 1  + (−0.0117 − 0.0117i)5-s − 0.659·7-s + (0.487 − 0.487i)11-s + (−0.850 + 0.850i)13-s + 0.757·17-s + (0.138 + 0.138i)19-s + 0.712·23-s − 0.999i·25-s + (−0.898 + 0.898i)29-s − 0.652i·31-s + (0.00773 + 0.00773i)35-s + (−1.11 − 1.11i)37-s − 0.0804i·41-s + (−0.0183 + 0.0183i)43-s − 1.69i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.548 + 0.835i$
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: \((2,\ 1152,\ (\ :1),\ -0.548 + 0.835i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8014144907\)
\(L(\frac12)\) \(\approx\) \(0.8014144907\)
\(L(2)\) 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 \( 1 + (0.0586 + 0.0586i)T + 25iT^{2} \)
7 \( 1 + 4.61T + 49T^{2} \)
11 \( 1 + (-5.36 + 5.36i)T - 121iT^{2} \)
13 \( 1 + (11.0 - 11.0i)T - 169iT^{2} \)
17 \( 1 - 12.8T + 289T^{2} \)
19 \( 1 + (-2.63 - 2.63i)T + 361iT^{2} \)
23 \( 1 - 16.3T + 529T^{2} \)
29 \( 1 + (26.0 - 26.0i)T - 841iT^{2} \)
31 \( 1 + 20.2iT - 961T^{2} \)
37 \( 1 + (41.2 + 41.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 3.29iT - 1.68e3T^{2} \)
43 \( 1 + (0.786 - 0.786i)T - 1.84e3iT^{2} \)
47 \( 1 + 79.7iT - 2.20e3T^{2} \)
53 \( 1 + (-1.06 - 1.06i)T + 2.80e3iT^{2} \)
59 \( 1 + (32.5 - 32.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (15.2 - 15.2i)T - 3.72e3iT^{2} \)
67 \( 1 + (60.0 + 60.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 56.3T + 5.04e3T^{2} \)
73 \( 1 + 9.70iT - 5.32e3T^{2} \)
79 \( 1 + 84.4iT - 6.24e3T^{2} \)
83 \( 1 + (26.7 + 26.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 115. iT - 7.92e3T^{2} \)
97 \( 1 + 146.T + 9.40e3T^{2} \)
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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

−9.299407868172990717369171712077, −8.698492982885657171732421914743, −7.52635332443795249596097798022, −6.87626499233774983718254721123, −5.98640472822587813968829562688, −5.06462178873551776911914956996, −3.94927002705153903519685234139, −3.08262745336625497548507604340, −1.80046093408871849594241342659, −0.24586260173011279235043628546, 1.30971585975397005246660388110, 2.78720011658679223310657015287, 3.57838372346465538282961829297, 4.82635575658603751992454383208, 5.59924099785520371380328792569, 6.65231006469999583384282488081, 7.37338027671999969950543843549, 8.156052755819975005049865575721, 9.359027908742283327165160862738, 9.703295576016063679519748608806

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