Properties

Label 2-1152-96.59-c1-0-9
Degree $2$
Conductor $1152$
Sign $0.881 + 0.472i$
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  + (0.0963 − 0.232i)5-s + (−0.617 + 0.617i)7-s + (0.505 − 1.21i)11-s + (3.41 − 1.41i)13-s − 2.76·17-s + (−0.189 − 0.458i)19-s + (4.46 − 4.46i)23-s + (3.49 + 3.49i)25-s + (0.0101 − 0.00418i)29-s − 4.03i·31-s + (0.0841 + 0.203i)35-s + (6.30 + 2.61i)37-s + (−5.34 − 5.34i)41-s + (10.1 + 4.21i)43-s − 11.5i·47-s + ⋯
L(s)  = 1  + (0.0430 − 0.104i)5-s + (−0.233 + 0.233i)7-s + (0.152 − 0.367i)11-s + (0.947 − 0.392i)13-s − 0.669·17-s + (−0.0435 − 0.105i)19-s + (0.931 − 0.931i)23-s + (0.698 + 0.698i)25-s + (0.00187 − 0.000777i)29-s − 0.724i·31-s + (0.0142 + 0.0343i)35-s + (1.03 + 0.429i)37-s + (−0.834 − 0.834i)41-s + (1.55 + 0.642i)43-s − 1.69i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.881 + 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.642079291\)
\(L(\frac12)\) \(\approx\) \(1.642079291\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0963 + 0.232i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.617 - 0.617i)T - 7iT^{2} \)
11 \( 1 + (-0.505 + 1.21i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-3.41 + 1.41i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 2.76T + 17T^{2} \)
19 \( 1 + (0.189 + 0.458i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.46 + 4.46i)T - 23iT^{2} \)
29 \( 1 + (-0.0101 + 0.00418i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 4.03iT - 31T^{2} \)
37 \( 1 + (-6.30 - 2.61i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.34 + 5.34i)T + 41iT^{2} \)
43 \( 1 + (-10.1 - 4.21i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 11.5iT - 47T^{2} \)
53 \( 1 + (-9.04 - 3.74i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.939 - 0.389i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (2.97 + 7.17i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-7.40 + 3.06i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (1.20 + 1.20i)T + 71iT^{2} \)
73 \( 1 + (-3.73 + 3.73i)T - 73iT^{2} \)
79 \( 1 - 7.22T + 79T^{2} \)
83 \( 1 + (11.2 - 4.67i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-3.70 + 3.70i)T - 89iT^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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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

−9.600664382794263401417850591806, −8.865959252747827669862749730197, −8.296741788693857543337187152659, −7.16601622877916482420599383555, −6.35758798349769913079091241400, −5.56725120440975150650435969981, −4.52324212444968277278458547163, −3.48224740268120191201870289735, −2.44296985265720127789438518880, −0.878606765266000265737848460170, 1.18701140025449478612751853184, 2.58343668614243816120968963052, 3.73594348056264213863138283068, 4.58611739442469800604533506449, 5.70001524328496117147644923923, 6.62686340018352911250305551764, 7.22357763511192960763097081550, 8.339134263921078098879350376174, 9.050433616675813180141980585598, 9.802770941196098180036052837338

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