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Properties

Label 2-1152-72.59-c1-0-23
Degree $2$
Conductor $1152$
Sign $-0.408 + 0.912i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 − 1.10i)3-s + (−1.43 + 2.49i)5-s + (−1.90 + 1.09i)7-s + (0.565 + 2.94i)9-s + (−5.42 + 3.13i)11-s + (4.45 + 2.57i)13-s + (4.67 − 1.74i)15-s − 5.43i·17-s − 2.07·19-s + (3.75 + 0.632i)21-s + (2.72 − 4.72i)23-s + (−1.64 − 2.84i)25-s + (2.49 − 4.55i)27-s + (−4.80 − 8.31i)29-s + (2.60 + 1.50i)31-s + ⋯
L(s)  = 1  + (−0.770 − 0.636i)3-s + (−0.643 + 1.11i)5-s + (−0.719 + 0.415i)7-s + (0.188 + 0.982i)9-s + (−1.63 + 0.945i)11-s + (1.23 + 0.713i)13-s + (1.20 − 0.449i)15-s − 1.31i·17-s − 0.476·19-s + (0.819 + 0.138i)21-s + (0.569 − 0.985i)23-s + (−0.328 − 0.568i)25-s + (0.480 − 0.877i)27-s + (−0.891 − 1.54i)29-s + (0.468 + 0.270i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.408 + 0.912i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2202790302\)
\(L(\frac12)\) \(\approx\) \(0.2202790302\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 + 1.10i)T \)
good5 \( 1 + (1.43 - 2.49i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.90 - 1.09i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.42 - 3.13i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.45 - 2.57i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.43iT - 17T^{2} \)
19 \( 1 + 2.07T + 19T^{2} \)
23 \( 1 + (-2.72 + 4.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.80 + 8.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.60 - 1.50i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.08iT - 37T^{2} \)
41 \( 1 + (2.92 + 1.68i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.10 + 7.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.381 + 0.660i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.54T + 53T^{2} \)
59 \( 1 + (-1.77 - 1.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.32 + 0.763i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.819 - 1.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.97T + 71T^{2} \)
73 \( 1 - 1.45T + 73T^{2} \)
79 \( 1 + (-5.99 + 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.70 + 0.985i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.84iT - 89T^{2} \)
97 \( 1 + (6.56 + 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.803797163100484389614869368702, −8.523126132480540124529161785806, −7.65450565426917950888911198722, −6.91893459848003575450743121692, −6.43792430993860999361007124212, −5.38663554820952024889042259947, −4.39866330784396062985067209261, −3.01913388394138646779111991362, −2.21732229361032507545551990492, −0.12710371866403679459960416242, 1.03644853568899298035916794630, 3.34670183072273888456178069478, 3.84997550161614987293714445456, 5.08651444396830036768009012922, 5.62274053282819404467069831546, 6.49045091636691268356967558893, 7.81602064030463396268815598384, 8.445618073673459405592704815950, 9.185814586216845403394986898689, 10.31375792678680761269589224015

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