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Properties

Label 2-1152-16.3-c2-0-36
Degree $2$
Conductor $1152$
Sign $-0.941 - 0.336i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.941 - 0.336i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.941 - 0.336i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2489899206\)
\(L(\frac12)\) \(\approx\) \(0.2489899206\)
\(L(2)\) 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

−9.218725269695437496406729674056, −8.619669561607852417579664763192, −7.48313911632455114908344732744, −6.32045359613838788548406072669, −5.93617929548832148247046944552, −4.84675996000904507003193182665, −3.98877991036925706387408485482, −2.54186896168182777207776760783, −1.56982033344848320066836653156, −0.06669333483787078729112138840, 1.96078996779616807960549601169, 2.78735596549997785748757408460, 3.70689988249547166547816234475, 5.05868890032856749409621661563, 6.17461396617750714119830053741, 6.74160734769468216541106743022, 7.05503698567816481179704022366, 8.845906832048179280217261981539, 9.226784290336402892052088379491, 10.03461368237164237741444073685

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