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Properties

Label 2-1152-96.59-c1-0-13
Degree $2$
Conductor $1152$
Sign $0.414 + 0.909i$
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  + (1.32 − 3.19i)5-s + (2.32 − 2.32i)7-s + (−1.47 + 3.55i)11-s + (4.49 − 1.86i)13-s + 4.93·17-s + (1.98 + 4.79i)19-s + (−1.08 + 1.08i)23-s + (−4.91 − 4.91i)25-s + (−3.43 + 1.42i)29-s − 8.82i·31-s + (−4.35 − 10.5i)35-s + (1.94 + 0.804i)37-s + (−5.87 − 5.87i)41-s + (2.44 + 1.01i)43-s + 1.61i·47-s + ⋯
L(s)  = 1  + (0.591 − 1.42i)5-s + (0.880 − 0.880i)7-s + (−0.443 + 1.07i)11-s + (1.24 − 0.516i)13-s + 1.19·17-s + (0.455 + 1.09i)19-s + (−0.227 + 0.227i)23-s + (−0.982 − 0.982i)25-s + (−0.637 + 0.263i)29-s − 1.58i·31-s + (−0.736 − 1.77i)35-s + (0.319 + 0.132i)37-s + (−0.917 − 0.917i)41-s + (0.372 + 0.154i)43-s + 0.236i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.414 + 0.909i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.414 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.106380615\)
\(L(\frac12)\) \(\approx\) \(2.106380615\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 3.19i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.32 + 2.32i)T - 7iT^{2} \)
11 \( 1 + (1.47 - 3.55i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-4.49 + 1.86i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 4.93T + 17T^{2} \)
19 \( 1 + (-1.98 - 4.79i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.08 - 1.08i)T - 23iT^{2} \)
29 \( 1 + (3.43 - 1.42i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 8.82iT - 31T^{2} \)
37 \( 1 + (-1.94 - 0.804i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.87 + 5.87i)T + 41iT^{2} \)
43 \( 1 + (-2.44 - 1.01i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 1.61iT - 47T^{2} \)
53 \( 1 + (5.62 + 2.32i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (7.67 + 3.17i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-3.16 - 7.65i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.31 - 1.37i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (2.13 + 2.13i)T + 71iT^{2} \)
73 \( 1 + (1.81 - 1.81i)T - 73iT^{2} \)
79 \( 1 + 1.42T + 79T^{2} \)
83 \( 1 + (1.04 - 0.431i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.708 - 0.708i)T - 89iT^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−9.764489598892082611870302084720, −8.794947126998766463071040887692, −7.87034564416355799932501994913, −7.58937763454423836858327373663, −5.97587868856679359978434325450, −5.37644907281700188044597155581, −4.52036491521862182674483369932, −3.65211178179520080145019439181, −1.79435897777319581041888132292, −1.07680634203583795683621704861, 1.56245541525318523643392540175, 2.79121002091539978835891763525, 3.43515061186510278567794820088, 5.01999530229848435866591177203, 5.84992082916003850511839671657, 6.44200379619761046449617166099, 7.47275858182480703946981342450, 8.367921069616973377820366111822, 9.050291580355751436208665649583, 10.06415161161999737470764443523

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