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Properties

Label 2-1152-96.11-c1-0-2
Degree $2$
Conductor $1152$
Sign $-0.0619 - 0.998i$
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  + (−2.97 − 1.23i)5-s + (−0.237 + 0.237i)7-s + (2.12 + 0.879i)11-s + (0.0390 + 0.0943i)13-s − 4.16·17-s + (4.25 − 1.76i)19-s + (−4.84 + 4.84i)23-s + (3.79 + 3.79i)25-s + (−2.90 − 7.01i)29-s + 9.88i·31-s + (0.999 − 0.414i)35-s + (0.175 − 0.424i)37-s + (7.67 + 7.67i)41-s + (−2.99 + 7.23i)43-s + 6.10i·47-s + ⋯
L(s)  = 1  + (−1.33 − 0.550i)5-s + (−0.0898 + 0.0898i)7-s + (0.640 + 0.265i)11-s + (0.0108 + 0.0261i)13-s − 1.00·17-s + (0.975 − 0.404i)19-s + (−1.01 + 1.01i)23-s + (0.758 + 0.758i)25-s + (−0.539 − 1.30i)29-s + 1.77i·31-s + (0.169 − 0.0700i)35-s + (0.0288 − 0.0697i)37-s + (1.19 + 1.19i)41-s + (−0.456 + 1.10i)43-s + 0.891i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0619 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0619 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.0619 - 0.998i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.0619 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7196589328\)
\(L(\frac12)\) \(\approx\) \(0.7196589328\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.97 + 1.23i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.237 - 0.237i)T - 7iT^{2} \)
11 \( 1 + (-2.12 - 0.879i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.0390 - 0.0943i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 4.16T + 17T^{2} \)
19 \( 1 + (-4.25 + 1.76i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.84 - 4.84i)T - 23iT^{2} \)
29 \( 1 + (2.90 + 7.01i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 9.88iT - 31T^{2} \)
37 \( 1 + (-0.175 + 0.424i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-7.67 - 7.67i)T + 41iT^{2} \)
43 \( 1 + (2.99 - 7.23i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 6.10iT - 47T^{2} \)
53 \( 1 + (4.28 - 10.3i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.19 + 2.88i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-6.29 + 2.60i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-5.66 - 13.6i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-3.49 - 3.49i)T + 71iT^{2} \)
73 \( 1 + (1.42 - 1.42i)T - 73iT^{2} \)
79 \( 1 + 1.53T + 79T^{2} \)
83 \( 1 + (2.95 + 7.14i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-1.92 + 1.92i)T - 89iT^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−9.769575883154950650176290895642, −9.223252861558371396976765297574, −8.265933238745930302850682040871, −7.65852250528276827335568014835, −6.82456022025814757768251602263, −5.77344024669472671587533847790, −4.57538265427541980622165603525, −4.06470026536977112571519111294, −2.93437774519598454685807574571, −1.28579348256318553622071391425, 0.34240183523661388658348542484, 2.23637367133706373004311621827, 3.65297669327926913416963725744, 3.97654426478449053102777830921, 5.27436583407286317021990315847, 6.44407850340375433024463535756, 7.11813545430393548602015862050, 7.913236252388197379275759720550, 8.648927127326529813628715657088, 9.562352331005430789867743183381

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