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Properties

Label 2-1152-72.59-c1-0-27
Degree $2$
Conductor $1152$
Sign $0.874 - 0.485i$
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  + (1.68 + 0.388i)3-s + (−0.538 + 0.932i)5-s + (4.07 − 2.35i)7-s + (2.69 + 1.31i)9-s + (−4.28 + 2.47i)11-s + (1.05 + 0.606i)13-s + (−1.27 + 1.36i)15-s + 3.12i·17-s + 2.05·19-s + (7.78 − 2.38i)21-s + (1.71 − 2.97i)23-s + (1.91 + 3.32i)25-s + (4.04 + 3.26i)27-s + (−0.723 − 1.25i)29-s + (6.45 + 3.72i)31-s + ⋯
L(s)  = 1  + (0.974 + 0.224i)3-s + (−0.240 + 0.417i)5-s + (1.53 − 0.888i)7-s + (0.899 + 0.437i)9-s + (−1.29 + 0.746i)11-s + (0.291 + 0.168i)13-s + (−0.328 + 0.352i)15-s + 0.756i·17-s + 0.472·19-s + (1.69 − 0.520i)21-s + (0.358 − 0.620i)23-s + (0.383 + 0.665i)25-s + (0.777 + 0.628i)27-s + (−0.134 − 0.232i)29-s + (1.15 + 0.669i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.565684216\)
\(L(\frac12)\) \(\approx\) \(2.565684216\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.388i)T \)
good5 \( 1 + (0.538 - 0.932i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-4.07 + 2.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.28 - 2.47i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.05 - 0.606i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.12iT - 17T^{2} \)
19 \( 1 - 2.05T + 19T^{2} \)
23 \( 1 + (-1.71 + 2.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.723 + 1.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.45 - 3.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.04iT - 37T^{2} \)
41 \( 1 + (8.75 + 5.05i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0592 - 0.102i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.98 + 3.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.401T + 53T^{2} \)
59 \( 1 + (-8.97 - 5.18i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.7 - 6.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.94 - 8.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 0.327T + 73T^{2} \)
79 \( 1 + (8.98 - 5.18i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.21 + 4.74i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.80iT - 89T^{2} \)
97 \( 1 + (-2.79 - 4.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.12032537472265553355846284274, −8.798417715853049455423955467993, −8.198847968332721152890558212714, −7.47685404781366087309586424515, −7.00581379664307668131783206471, −5.29575261694105862131522778735, −4.56869018223684686608555217056, −3.73727083959708087243568470369, −2.54229641485953976338523275006, −1.49456435130998762810398497819, 1.19782497250056022933582039932, 2.43982594302024268276537323393, 3.24624420024989939488456106480, 4.77637643481416267951769727825, 5.14240966341129951683895805794, 6.42516692632423077067847304666, 7.83308657245303455594235337399, 8.025817063025445917309079587093, 8.661255836998725653733120618501, 9.515562279736488634383514717889

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