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Properties

Label 2-1152-96.35-c1-0-3
Degree $2$
Conductor $1152$
Sign $-0.190 - 0.981i$
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  + (−1.65 + 0.686i)5-s + (0.456 + 0.456i)7-s + (2.79 − 1.15i)11-s + (−1.29 + 3.12i)13-s − 3.55·17-s + (5.61 + 2.32i)19-s + (−4.10 − 4.10i)23-s + (−1.26 + 1.26i)25-s + (−1.87 + 4.53i)29-s + 0.580i·31-s + (−1.06 − 0.442i)35-s + (2.58 + 6.22i)37-s + (−2.98 + 2.98i)41-s + (2.78 + 6.72i)43-s + 8.67i·47-s + ⋯
L(s)  = 1  + (−0.740 + 0.306i)5-s + (0.172 + 0.172i)7-s + (0.841 − 0.348i)11-s + (−0.359 + 0.867i)13-s − 0.862·17-s + (1.28 + 0.533i)19-s + (−0.855 − 0.855i)23-s + (−0.252 + 0.252i)25-s + (−0.348 + 0.841i)29-s + 0.104i·31-s + (−0.180 − 0.0748i)35-s + (0.424 + 1.02i)37-s + (−0.465 + 0.465i)41-s + (0.424 + 1.02i)43-s + 1.26i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.190 - 0.981i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.066974043\)
\(L(\frac12)\) \(\approx\) \(1.066974043\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 - 0.686i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.456 - 0.456i)T + 7iT^{2} \)
11 \( 1 + (-2.79 + 1.15i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.29 - 3.12i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.55T + 17T^{2} \)
19 \( 1 + (-5.61 - 2.32i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.10 + 4.10i)T + 23iT^{2} \)
29 \( 1 + (1.87 - 4.53i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 0.580iT - 31T^{2} \)
37 \( 1 + (-2.58 - 6.22i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.98 - 2.98i)T - 41iT^{2} \)
43 \( 1 + (-2.78 - 6.72i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 8.67iT - 47T^{2} \)
53 \( 1 + (-3.70 - 8.95i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.77 + 4.28i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (9.49 + 3.93i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.50 - 8.46i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (7.84 - 7.84i)T - 71iT^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 - 5.27T + 79T^{2} \)
83 \( 1 + (1.80 - 4.36i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (12.4 + 12.4i)T + 89iT^{2} \)
97 \( 1 - 9.99T + 97T^{2} \)
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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

−9.922781559367443301762066557841, −9.204604429999515474550337727566, −8.381347147064647992274980041573, −7.53994885651753025402458905650, −6.75598400139025305166782061367, −5.93811752902740749582874209602, −4.68763273407654855544981208163, −3.94145557131151826820134282722, −2.90442179277840540750297466745, −1.48224681412066836896729948719, 0.48152734240696795552566164715, 2.06277559480265178037092197920, 3.49593108142451602935391666042, 4.25475766547389724583160537583, 5.21313478923361491293976029754, 6.19116488286434946592212085032, 7.39365349146296720180843661212, 7.70471435246058024670079599300, 8.805166846691404327370791352482, 9.493247547363304147421619406968

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