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Properties

Label 2-1152-48.35-c3-0-40
Degree $2$
Conductor $1152$
Sign $0.196 + 0.980i$
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  + (2.40 + 2.40i)5-s + 11.7·7-s + (34.7 − 34.7i)11-s + (3.17 + 3.17i)13-s − 98.0i·17-s + (−15.9 + 15.9i)19-s + 69.6i·23-s − 113. i·25-s + (−15.9 + 15.9i)29-s − 121. i·31-s + (28.2 + 28.2i)35-s + (37.0 − 37.0i)37-s − 59.3·41-s + (−241. − 241. i)43-s − 395.·47-s + ⋯
L(s)  = 1  + (0.215 + 0.215i)5-s + 0.632·7-s + (0.952 − 0.952i)11-s + (0.0678 + 0.0678i)13-s − 1.39i·17-s + (−0.192 + 0.192i)19-s + 0.631i·23-s − 0.907i·25-s + (−0.102 + 0.102i)29-s − 0.702i·31-s + (0.136 + 0.136i)35-s + (0.164 − 0.164i)37-s − 0.225·41-s + (−0.857 − 0.857i)43-s − 1.22·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.196 + 0.980i$
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: \((2,\ 1152,\ (\ :3/2),\ 0.196 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.159935611\)
\(L(\frac12)\) \(\approx\) \(2.159935611\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.40 - 2.40i)T + 125iT^{2} \)
7 \( 1 - 11.7T + 343T^{2} \)
11 \( 1 + (-34.7 + 34.7i)T - 1.33e3iT^{2} \)
13 \( 1 + (-3.17 - 3.17i)T + 2.19e3iT^{2} \)
17 \( 1 + 98.0iT - 4.91e3T^{2} \)
19 \( 1 + (15.9 - 15.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 69.6iT - 1.21e4T^{2} \)
29 \( 1 + (15.9 - 15.9i)T - 2.43e4iT^{2} \)
31 \( 1 + 121. iT - 2.97e4T^{2} \)
37 \( 1 + (-37.0 + 37.0i)T - 5.06e4iT^{2} \)
41 \( 1 + 59.3T + 6.89e4T^{2} \)
43 \( 1 + (241. + 241. i)T + 7.95e4iT^{2} \)
47 \( 1 + 395.T + 1.03e5T^{2} \)
53 \( 1 + (-458. - 458. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-257. + 257. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-373. - 373. i)T + 2.26e5iT^{2} \)
67 \( 1 + (648. - 648. i)T - 3.00e5iT^{2} \)
71 \( 1 + 787. iT - 3.57e5T^{2} \)
73 \( 1 + 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 + 382. iT - 4.93e5T^{2} \)
83 \( 1 + (-491. - 491. i)T + 5.71e5iT^{2} \)
89 \( 1 + 624.T + 7.04e5T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−9.191768384846535453613118530271, −8.497247119175090048488897446165, −7.60319257162506814578895515465, −6.70645327538099041503723117429, −5.89518736703641490513638100286, −4.97145508929735485193944596890, −3.95776051662457719189895892791, −2.93549083780156753406426296175, −1.72867137715572084999014489134, −0.53041947632673942795551951663, 1.28498501373589060273668294185, 2.01788951336984279263756292479, 3.53190119491204669582319857893, 4.44951653530567887740275545160, 5.24269232840489468756398517248, 6.35552174215838897905844084041, 7.01256182445476910022081182274, 8.130409635233732002822039808954, 8.686215934935804051802684277615, 9.646224252213889226219763002607

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