Properties

Degree $2$
Conductor $1152$
Sign $0.668 - 0.744i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.49 + 6.49i)5-s + 3.94·7-s + (4.31 − 4.31i)11-s + (−4.06 + 4.06i)13-s + 14.5·17-s + (−4.94 − 4.94i)19-s + 43.6·23-s + 59.3i·25-s + (25.0 − 25.0i)29-s − 32.5i·31-s + (25.6 + 25.6i)35-s + (−4.14 − 4.14i)37-s + 55.3i·41-s + (16.1 − 16.1i)43-s − 7.92i·47-s + ⋯
L(s)  = 1  + (1.29 + 1.29i)5-s + 0.563·7-s + (0.391 − 0.391i)11-s + (−0.312 + 0.312i)13-s + 0.856·17-s + (−0.260 − 0.260i)19-s + 1.89·23-s + 2.37i·25-s + (0.865 − 0.865i)29-s − 1.04i·31-s + (0.731 + 0.731i)35-s + (−0.111 − 0.111i)37-s + 1.34i·41-s + (0.374 − 0.374i)43-s − 0.168i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.744i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.668 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.668 - 0.744i$
Motivic weight: \(2\)
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.668 - 0.744i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.912325462\)
\(L(\frac12)\) \(\approx\) \(2.912325462\)
\(L(2)\) 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 \( 1 + (-6.49 - 6.49i)T + 25iT^{2} \)
7 \( 1 - 3.94T + 49T^{2} \)
11 \( 1 + (-4.31 + 4.31i)T - 121iT^{2} \)
13 \( 1 + (4.06 - 4.06i)T - 169iT^{2} \)
17 \( 1 - 14.5T + 289T^{2} \)
19 \( 1 + (4.94 + 4.94i)T + 361iT^{2} \)
23 \( 1 - 43.6T + 529T^{2} \)
29 \( 1 + (-25.0 + 25.0i)T - 841iT^{2} \)
31 \( 1 + 32.5iT - 961T^{2} \)
37 \( 1 + (4.14 + 4.14i)T + 1.36e3iT^{2} \)
41 \( 1 - 55.3iT - 1.68e3T^{2} \)
43 \( 1 + (-16.1 + 16.1i)T - 1.84e3iT^{2} \)
47 \( 1 + 7.92iT - 2.20e3T^{2} \)
53 \( 1 + (31.5 + 31.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (49.7 - 49.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (44.4 - 44.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (-1.64 - 1.64i)T + 4.48e3iT^{2} \)
71 \( 1 + 24.1T + 5.04e3T^{2} \)
73 \( 1 + 10.7iT - 5.32e3T^{2} \)
79 \( 1 - 72.0iT - 6.24e3T^{2} \)
83 \( 1 + (-42.0 - 42.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 28.9iT - 7.92e3T^{2} \)
97 \( 1 + 54.2T + 9.40e3T^{2} \)
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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

−9.716140815757360921859535765528, −9.142703790862654836470848119921, −7.988763799228724660839967302896, −7.06965190560486436523108926893, −6.38127960011087827853932798180, −5.63809889600195601443308652955, −4.63199798501540958820856121813, −3.21615675569831966828054699581, −2.46441972804747479869510101783, −1.29022636573418395961156402056, 1.03690570340037813871166360273, 1.75673183587893773247340692467, 3.11098955818444672934154537045, 4.73869935565173222039092874597, 5.03290184352462249205549711402, 5.96216479642774398896717856563, 6.95199795949824635983503271122, 8.022213810867759033835597666178, 8.885215464801214972240359102140, 9.325792992982894336781545533669

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