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Properties

Label 2-1152-96.35-c1-0-0
Degree $2$
Conductor $1152$
Sign $-0.916 - 0.399i$
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  + (1.64 − 0.682i)5-s + (−2.51 − 2.51i)7-s + (−2.69 + 1.11i)11-s + (−1.76 + 4.25i)13-s − 6.10·17-s + (−3.43 − 1.42i)19-s + (−0.525 − 0.525i)23-s + (−1.28 + 1.28i)25-s + (1.46 − 3.53i)29-s + 7.55i·31-s + (−5.85 − 2.42i)35-s + (2.30 + 5.56i)37-s + (3.04 − 3.04i)41-s + (3.31 + 8.00i)43-s + 8.59i·47-s + ⋯
L(s)  = 1  + (0.737 − 0.305i)5-s + (−0.949 − 0.949i)7-s + (−0.811 + 0.336i)11-s + (−0.488 + 1.17i)13-s − 1.47·17-s + (−0.787 − 0.326i)19-s + (−0.109 − 0.109i)23-s + (−0.256 + 0.256i)25-s + (0.272 − 0.656i)29-s + 1.35i·31-s + (−0.990 − 0.410i)35-s + (0.378 + 0.914i)37-s + (0.475 − 0.475i)41-s + (0.505 + 1.22i)43-s + 1.25i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.916 - 0.399i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.916 - 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1078160180\)
\(L(\frac12)\) \(\approx\) \(0.1078160180\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 0.682i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.51 + 2.51i)T + 7iT^{2} \)
11 \( 1 + (2.69 - 1.11i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.76 - 4.25i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 6.10T + 17T^{2} \)
19 \( 1 + (3.43 + 1.42i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.525 + 0.525i)T + 23iT^{2} \)
29 \( 1 + (-1.46 + 3.53i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 7.55iT - 31T^{2} \)
37 \( 1 + (-2.30 - 5.56i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.04 + 3.04i)T - 41iT^{2} \)
43 \( 1 + (-3.31 - 8.00i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 8.59iT - 47T^{2} \)
53 \( 1 + (3.78 + 9.14i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.73 + 9.01i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.41 - 1.41i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (0.0538 - 0.130i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (1.31 - 1.31i)T - 71iT^{2} \)
73 \( 1 + (10.9 + 10.9i)T + 73iT^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (-4.38 + 10.5i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.97 + 5.97i)T + 89iT^{2} \)
97 \( 1 + 3.21T + 97T^{2} \)
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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

−10.03024383208713679305201457633, −9.457466927663757536827016998473, −8.703493779944950251733374447331, −7.54704874354671404010333669261, −6.69730377257528075731048094678, −6.21274119554667313788487575864, −4.81237482803354733193860900539, −4.26876148755637126430157054570, −2.85324507015002940366962374430, −1.80242248200105688039162434831, 0.04166392686687053106278310055, 2.35239434693976971673587698389, 2.73181790151061910907677970449, 4.14733204301194160701776925599, 5.53112560056857537512678472215, 5.89635565990946126064229860459, 6.78821463256583259007822303293, 7.85429723979613120434340289556, 8.759555097531386212823798467403, 9.438885761201487741505958165694

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