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Properties

Label 2-1152-96.83-c1-0-1
Degree $2$
Conductor $1152$
Sign $-0.137 - 0.990i$
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  + (1.12 + 2.71i)5-s + (−3.03 − 3.03i)7-s + (0.616 + 1.48i)11-s + (3.35 + 1.38i)13-s − 4.76·17-s + (−1.13 + 2.73i)19-s + (4.11 + 4.11i)23-s + (−2.55 + 2.55i)25-s + (8.16 + 3.38i)29-s + 6.16i·31-s + (4.81 − 11.6i)35-s + (−9.38 + 3.88i)37-s + (0.169 − 0.169i)41-s + (−7.57 + 3.13i)43-s + 2.44i·47-s + ⋯
L(s)  = 1  + (0.502 + 1.21i)5-s + (−1.14 − 1.14i)7-s + (0.185 + 0.449i)11-s + (0.929 + 0.385i)13-s − 1.15·17-s + (−0.259 + 0.627i)19-s + (0.857 + 0.857i)23-s + (−0.511 + 0.511i)25-s + (1.51 + 0.628i)29-s + 1.10i·31-s + (0.814 − 1.96i)35-s + (−1.54 + 0.639i)37-s + (0.0265 − 0.0265i)41-s + (−1.15 + 0.478i)43-s + 0.355i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.137 - 0.990i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.137 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.278064517\)
\(L(\frac12)\) \(\approx\) \(1.278064517\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 2.71i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (3.03 + 3.03i)T + 7iT^{2} \)
11 \( 1 + (-0.616 - 1.48i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-3.35 - 1.38i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 4.76T + 17T^{2} \)
19 \( 1 + (1.13 - 2.73i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.11 - 4.11i)T + 23iT^{2} \)
29 \( 1 + (-8.16 - 3.38i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 + (9.38 - 3.88i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.169 + 0.169i)T - 41iT^{2} \)
43 \( 1 + (7.57 - 3.13i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 2.44iT - 47T^{2} \)
53 \( 1 + (1.99 - 0.824i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.21 + 0.503i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-1.04 + 2.52i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (3.91 + 1.62i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (5.28 - 5.28i)T - 71iT^{2} \)
73 \( 1 + (1.57 + 1.57i)T + 73iT^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + (-3.31 - 1.37i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (6.99 + 6.99i)T + 89iT^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−10.27068112339904580348959213882, −9.356644786171868302366260139154, −8.464990520035725974680804236123, −7.09006081199192662130158767617, −6.75567000688003849426473879011, −6.23373704455275756065428019370, −4.77021462395367293746301940504, −3.62210129396617399131352648310, −3.02525438056124115437105415440, −1.53364940349144543143533886522, 0.55937009626138170340551522414, 2.15385641596630967693268528531, 3.19937974756054441674166305506, 4.47921869377534462292111409489, 5.39408861650352741768193228254, 6.17229477477050894794380333893, 6.77225150206357590413165646310, 8.507919212144355523772218806671, 8.695385637118329281246178112762, 9.324291322965652724857390457742

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