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Properties

Label 2-1152-96.59-c1-0-15
Degree $2$
Conductor $1152$
Sign $-0.0542 + 0.998i$
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  + (0.352 − 0.852i)5-s + (3.43 − 3.43i)7-s + (1.44 − 3.49i)11-s + (−0.258 + 0.107i)13-s − 5.30·17-s + (−2.72 − 6.57i)19-s + (−2.23 + 2.23i)23-s + (2.93 + 2.93i)25-s + (−3.16 + 1.31i)29-s + 3.46i·31-s + (−1.71 − 4.13i)35-s + (1.27 + 0.528i)37-s + (5.28 + 5.28i)41-s + (−2.46 − 1.02i)43-s + 0.423i·47-s + ⋯
L(s)  = 1  + (0.157 − 0.381i)5-s + (1.29 − 1.29i)7-s + (0.436 − 1.05i)11-s + (−0.0717 + 0.0297i)13-s − 1.28·17-s + (−0.625 − 1.50i)19-s + (−0.466 + 0.466i)23-s + (0.586 + 0.586i)25-s + (−0.588 + 0.243i)29-s + 0.622i·31-s + (−0.289 − 0.699i)35-s + (0.209 + 0.0868i)37-s + (0.824 + 0.824i)41-s + (−0.376 − 0.155i)43-s + 0.0618i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0542 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0542 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.0542 + 0.998i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.0542 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.724569264\)
\(L(\frac12)\) \(\approx\) \(1.724569264\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.352 + 0.852i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.43 + 3.43i)T - 7iT^{2} \)
11 \( 1 + (-1.44 + 3.49i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.258 - 0.107i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 5.30T + 17T^{2} \)
19 \( 1 + (2.72 + 6.57i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.23 - 2.23i)T - 23iT^{2} \)
29 \( 1 + (3.16 - 1.31i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (-1.27 - 0.528i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.28 - 5.28i)T + 41iT^{2} \)
43 \( 1 + (2.46 + 1.02i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 0.423iT - 47T^{2} \)
53 \( 1 + (12.5 + 5.20i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-5.24 - 2.17i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.0138 - 0.0333i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-9.82 + 4.06i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (4.64 + 4.64i)T + 71iT^{2} \)
73 \( 1 + (-3.96 + 3.96i)T - 73iT^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (-0.867 + 0.359i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.82 + 4.82i)T - 89iT^{2} \)
97 \( 1 - 8.78T + 97T^{2} \)
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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

−9.428415687135706883440143028191, −8.739057100697517009187776052891, −8.019917896823959296035271298939, −7.10446159345975410841335146004, −6.35589534259066666846387844192, −5.02335520122249031099706618768, −4.52397244517274391333970350952, −3.47364718972581808842370031975, −1.92431339775133396022907655148, −0.75587789832848221279070782477, 1.89246780383775492381544788051, 2.35739311081653149310153287946, 4.07619842007228070376011431971, 4.80588515175609065761495984789, 5.84635129663790744029856940202, 6.55638393043979189819423261073, 7.68865594862699646907282932235, 8.381820253446996900793550910053, 9.117801198360068032300132909032, 9.977289837658620904511166866726

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