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Properties

Label 2-1152-96.35-c1-0-10
Degree $2$
Conductor $1152$
Sign $-0.229 + 0.973i$
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  + (−2.70 + 1.11i)5-s + (1.06 + 1.06i)7-s + (−5.29 + 2.19i)11-s + (1.67 − 4.04i)13-s + 3.44·17-s + (−3.23 − 1.33i)19-s + (−0.703 − 0.703i)23-s + (2.50 − 2.50i)25-s + (3.94 − 9.52i)29-s − 4.23i·31-s + (−4.07 − 1.68i)35-s + (−2.04 − 4.93i)37-s + (−3.53 + 3.53i)41-s + (3.38 + 8.16i)43-s − 4.33i·47-s + ⋯
L(s)  = 1  + (−1.20 + 0.500i)5-s + (0.403 + 0.403i)7-s + (−1.59 + 0.661i)11-s + (0.464 − 1.12i)13-s + 0.835·17-s + (−0.741 − 0.307i)19-s + (−0.146 − 0.146i)23-s + (0.501 − 0.501i)25-s + (0.732 − 1.76i)29-s − 0.761i·31-s + (−0.688 − 0.285i)35-s + (−0.336 − 0.812i)37-s + (−0.552 + 0.552i)41-s + (0.515 + 1.24i)43-s − 0.632i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5670668740\)
\(L(\frac12)\) \(\approx\) \(0.5670668740\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.70 - 1.11i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.06 - 1.06i)T + 7iT^{2} \)
11 \( 1 + (5.29 - 2.19i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.67 + 4.04i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 + (3.23 + 1.33i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.703 + 0.703i)T + 23iT^{2} \)
29 \( 1 + (-3.94 + 9.52i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 4.23iT - 31T^{2} \)
37 \( 1 + (2.04 + 4.93i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.53 - 3.53i)T - 41iT^{2} \)
43 \( 1 + (-3.38 - 8.16i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 4.33iT - 47T^{2} \)
53 \( 1 + (0.541 + 1.30i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.66 + 8.83i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.97 + 0.816i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.55 + 8.59i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.76 + 1.76i)T - 71iT^{2} \)
73 \( 1 + (1.16 + 1.16i)T + 73iT^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + (4.27 - 10.3i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (7.99 + 7.99i)T + 89iT^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−9.768600838040126799618211037978, −8.265011820273728933931597531461, −8.033380296119958244929979793245, −7.37539793966245103008192799057, −6.15671092130518308327691869748, −5.23144772676991430920069297813, −4.32632235236535380637088627692, −3.22946639855760969632344890570, −2.33194951532380845358854556464, −0.26098003952342040682584599859, 1.32052781484047955006220918031, 2.98905309214943565823740586340, 3.97075474394857649949475782854, 4.78732061493903448015506400223, 5.65242859987797440777468380998, 6.92918643393707684394722204813, 7.67480944310779213205034303722, 8.418373686959277669263453685167, 8.852325306119562512617570979748, 10.34382579431745317305467572989

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