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Properties

Label 2-1152-48.35-c3-0-6
Degree $2$
Conductor $1152$
Sign $-0.905 - 0.424i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.22 + 3.22i)5-s − 13.1·7-s + (−3.39 + 3.39i)11-s + (54.1 + 54.1i)13-s + 70.7i·17-s + (−32.5 + 32.5i)19-s − 16.4i·23-s − 104. i·25-s + (28.0 − 28.0i)29-s − 174. i·31-s + (−42.2 − 42.2i)35-s + (−116. + 116. i)37-s + 19.6·41-s + (−94.2 − 94.2i)43-s + 372.·47-s + ⋯
L(s)  = 1  + (0.288 + 0.288i)5-s − 0.707·7-s + (−0.0929 + 0.0929i)11-s + (1.15 + 1.15i)13-s + 1.00i·17-s + (−0.393 + 0.393i)19-s − 0.148i·23-s − 0.833i·25-s + (0.179 − 0.179i)29-s − 1.01i·31-s + (−0.203 − 0.203i)35-s + (−0.519 + 0.519i)37-s + 0.0747·41-s + (−0.334 − 0.334i)43-s + 1.15·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.905 - 0.424i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -0.905 - 0.424i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9135057134\)
\(L(\frac12)\) \(\approx\) \(0.9135057134\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.22 - 3.22i)T + 125iT^{2} \)
7 \( 1 + 13.1T + 343T^{2} \)
11 \( 1 + (3.39 - 3.39i)T - 1.33e3iT^{2} \)
13 \( 1 + (-54.1 - 54.1i)T + 2.19e3iT^{2} \)
17 \( 1 - 70.7iT - 4.91e3T^{2} \)
19 \( 1 + (32.5 - 32.5i)T - 6.85e3iT^{2} \)
23 \( 1 + 16.4iT - 1.21e4T^{2} \)
29 \( 1 + (-28.0 + 28.0i)T - 2.43e4iT^{2} \)
31 \( 1 + 174. iT - 2.97e4T^{2} \)
37 \( 1 + (116. - 116. i)T - 5.06e4iT^{2} \)
41 \( 1 - 19.6T + 6.89e4T^{2} \)
43 \( 1 + (94.2 + 94.2i)T + 7.95e4iT^{2} \)
47 \( 1 - 372.T + 1.03e5T^{2} \)
53 \( 1 + (162. + 162. i)T + 1.48e5iT^{2} \)
59 \( 1 + (610. - 610. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-531. - 531. i)T + 2.26e5iT^{2} \)
67 \( 1 + (562. - 562. i)T - 3.00e5iT^{2} \)
71 \( 1 + 1.16e3iT - 3.57e5T^{2} \)
73 \( 1 - 308. iT - 3.89e5T^{2} \)
79 \( 1 - 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + (469. + 469. i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.53e3T + 7.04e5T^{2} \)
97 \( 1 + 139.T + 9.12e5T^{2} \)
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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

−9.873330216363852690284094398525, −8.894697316414043300437592749246, −8.313078495588459770043253220142, −7.14121347980174802250051725843, −6.28106300476509856474842953083, −5.91483819581853098272471287764, −4.38592815680409750692157114323, −3.71237688955564156567000887986, −2.49527439848964494345599503930, −1.41041364942079104110264787290, 0.22059861550375174712731234906, 1.37783561574903038881812782505, 2.86899318741613042469453364505, 3.56219509358454080497001170560, 4.89461913805852510257199322398, 5.66551846834353359772424491543, 6.50261514831624945717466842234, 7.38405391991699881261053570480, 8.372708403712185515353714267372, 9.088182358455416472105750207134

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