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Properties

Label 2-1152-72.59-c1-0-25
Degree $2$
Conductor $1152$
Sign $0.408 - 0.912i$
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  + (1.33 + 1.10i)3-s + (−1.43 + 2.49i)5-s + (1.90 − 1.09i)7-s + (0.565 + 2.94i)9-s + (5.42 − 3.13i)11-s + (4.45 + 2.57i)13-s + (−4.67 + 1.74i)15-s − 5.43i·17-s + 2.07·19-s + (3.75 + 0.632i)21-s + (−2.72 + 4.72i)23-s + (−1.64 − 2.84i)25-s + (−2.49 + 4.55i)27-s + (−4.80 − 8.31i)29-s + (−2.60 − 1.50i)31-s + ⋯
L(s)  = 1  + (0.770 + 0.636i)3-s + (−0.643 + 1.11i)5-s + (0.719 − 0.415i)7-s + (0.188 + 0.982i)9-s + (1.63 − 0.945i)11-s + (1.23 + 0.713i)13-s + (−1.20 + 0.449i)15-s − 1.31i·17-s + 0.476·19-s + (0.819 + 0.138i)21-s + (−0.569 + 0.985i)23-s + (−0.328 − 0.568i)25-s + (−0.480 + 0.877i)27-s + (−0.891 − 1.54i)29-s + (−0.468 − 0.270i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.408 - 0.912i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.350471707\)
\(L(\frac12)\) \(\approx\) \(2.350471707\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 1.10i)T \)
good5 \( 1 + (1.43 - 2.49i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.90 + 1.09i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.42 + 3.13i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.45 - 2.57i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.43iT - 17T^{2} \)
19 \( 1 - 2.07T + 19T^{2} \)
23 \( 1 + (2.72 - 4.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.80 + 8.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.60 + 1.50i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.08iT - 37T^{2} \)
41 \( 1 + (2.92 + 1.68i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.10 - 7.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.381 - 0.660i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.54T + 53T^{2} \)
59 \( 1 + (1.77 + 1.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.32 + 0.763i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.819 + 1.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.97T + 71T^{2} \)
73 \( 1 - 1.45T + 73T^{2} \)
79 \( 1 + (5.99 - 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.70 - 0.985i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.84iT - 89T^{2} \)
97 \( 1 + (6.56 + 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.750649457415720278875886732453, −9.223126830242331017004702037297, −8.284496841197572153633427560339, −7.57976509353112900713251287779, −6.78472262578904671069635212999, −5.77915015246619638295688484359, −4.31609612602800008399923913219, −3.77993106202260953864433128564, −3.03012942799223347730685618641, −1.49621754773983056182065492097, 1.17433226953537426488939982587, 1.86741161344831678930371300143, 3.67934831112930745461744776694, 4.11456197766237404920162061235, 5.38087671298373412477401118449, 6.39809141447155082893647796406, 7.35414282979954514732189301079, 8.214815264112016418938033647098, 8.776857849071720783775450905480, 9.129308411334491650766756880382

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