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Properties

Label 2-1152-96.11-c1-0-4
Degree $2$
Conductor $1152$
Sign $0.993 - 0.110i$
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  + (−4.01 − 1.66i)5-s + (−2.72 + 2.72i)7-s + (−2.48 − 1.02i)11-s + (−0.146 − 0.352i)13-s + 1.69·17-s + (3.86 − 1.60i)19-s + (3.96 − 3.96i)23-s + (9.84 + 9.84i)25-s + (0.582 + 1.40i)29-s + 0.247i·31-s + (15.4 − 6.41i)35-s + (−2.88 + 6.95i)37-s + (4.26 + 4.26i)41-s + (1.21 − 2.92i)43-s + 8.76i·47-s + ⋯
L(s)  = 1  + (−1.79 − 0.744i)5-s + (−1.02 + 1.02i)7-s + (−0.748 − 0.309i)11-s + (−0.0405 − 0.0978i)13-s + 0.411·17-s + (0.887 − 0.367i)19-s + (0.827 − 0.827i)23-s + (1.96 + 1.96i)25-s + (0.108 + 0.261i)29-s + 0.0443i·31-s + (2.61 − 1.08i)35-s + (−0.473 + 1.14i)37-s + (0.666 + 0.666i)41-s + (0.184 − 0.445i)43-s + 1.27i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7800732080\)
\(L(\frac12)\) \(\approx\) \(0.7800732080\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (4.01 + 1.66i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.72 - 2.72i)T - 7iT^{2} \)
11 \( 1 + (2.48 + 1.02i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.146 + 0.352i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 1.69T + 17T^{2} \)
19 \( 1 + (-3.86 + 1.60i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.96 + 3.96i)T - 23iT^{2} \)
29 \( 1 + (-0.582 - 1.40i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 0.247iT - 31T^{2} \)
37 \( 1 + (2.88 - 6.95i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-4.26 - 4.26i)T + 41iT^{2} \)
43 \( 1 + (-1.21 + 2.92i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.76iT - 47T^{2} \)
53 \( 1 + (-3.22 + 7.78i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.77 + 4.28i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-6.31 + 2.61i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.346 + 0.837i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (9.14 + 9.14i)T + 71iT^{2} \)
73 \( 1 + (-0.0835 + 0.0835i)T - 73iT^{2} \)
79 \( 1 - 9.01T + 79T^{2} \)
83 \( 1 + (-2.10 - 5.07i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (3.77 - 3.77i)T - 89iT^{2} \)
97 \( 1 + 2.20T + 97T^{2} \)
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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

−9.620850819831755384049930668489, −8.860810648689827013820408495347, −8.235776310696612082190876707226, −7.49411972614528575031710441970, −6.55589375480764766023234971886, −5.36681944090608453495557708319, −4.71203566044349172027531482940, −3.44708468280203320147315472112, −2.87017948965011870594114971316, −0.69436981822891881199456281986, 0.60026762537725618264652867870, 2.87099440081096698504159961778, 3.59966900092014532122763960294, 4.24108673029291483550572369915, 5.52533468211711447722385676267, 6.85184146963031537687331815041, 7.36748384304840747820028772244, 7.76032107416215767335596364013, 8.926445064205405320483900309167, 10.04055005323878527070980067468

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