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Properties

Label 2-1152-48.11-c3-0-31
Degree $2$
Conductor $1152$
Sign $-0.377 + 0.925i$
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  + (−9.40 + 9.40i)5-s − 3.57·7-s + (−3.36 − 3.36i)11-s + (−26.9 + 26.9i)13-s − 12.7i·17-s + (50.0 + 50.0i)19-s + 208. i·23-s − 51.7i·25-s + (−134. − 134. i)29-s − 80.1i·31-s + (33.5 − 33.5i)35-s + (308. + 308. i)37-s − 172.·41-s + (87.0 − 87.0i)43-s − 525.·47-s + ⋯
L(s)  = 1  + (−0.840 + 0.840i)5-s − 0.192·7-s + (−0.0921 − 0.0921i)11-s + (−0.574 + 0.574i)13-s − 0.182i·17-s + (0.604 + 0.604i)19-s + 1.88i·23-s − 0.413i·25-s + (−0.859 − 0.859i)29-s − 0.464i·31-s + (0.162 − 0.162i)35-s + (1.37 + 1.37i)37-s − 0.658·41-s + (0.308 − 0.308i)43-s − 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.377 + 0.925i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -0.377 + 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.08782781899\)
\(L(\frac12)\) \(\approx\) \(0.08782781899\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.40 - 9.40i)T - 125iT^{2} \)
7 \( 1 + 3.57T + 343T^{2} \)
11 \( 1 + (3.36 + 3.36i)T + 1.33e3iT^{2} \)
13 \( 1 + (26.9 - 26.9i)T - 2.19e3iT^{2} \)
17 \( 1 + 12.7iT - 4.91e3T^{2} \)
19 \( 1 + (-50.0 - 50.0i)T + 6.85e3iT^{2} \)
23 \( 1 - 208. iT - 1.21e4T^{2} \)
29 \( 1 + (134. + 134. i)T + 2.43e4iT^{2} \)
31 \( 1 + 80.1iT - 2.97e4T^{2} \)
37 \( 1 + (-308. - 308. i)T + 5.06e4iT^{2} \)
41 \( 1 + 172.T + 6.89e4T^{2} \)
43 \( 1 + (-87.0 + 87.0i)T - 7.95e4iT^{2} \)
47 \( 1 + 525.T + 1.03e5T^{2} \)
53 \( 1 + (127. - 127. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-172. - 172. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-332. + 332. i)T - 2.26e5iT^{2} \)
67 \( 1 + (556. + 556. i)T + 3.00e5iT^{2} \)
71 \( 1 - 450. iT - 3.57e5T^{2} \)
73 \( 1 + 797. iT - 3.89e5T^{2} \)
79 \( 1 - 70.1iT - 4.93e5T^{2} \)
83 \( 1 + (-636. + 636. i)T - 5.71e5iT^{2} \)
89 \( 1 - 925.T + 7.04e5T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−9.473186749983721792862143394532, −8.009063355520370002557268356969, −7.61995936702421305309583106234, −6.77955208314095535285303685366, −5.86763966418603027690683953124, −4.77923849636758893236884940540, −3.69032021434584036600054712231, −3.04754191800740475974885634985, −1.69657362838656065199791738260, −0.02594300063446073703575797012, 0.909032624564623835862963186502, 2.46062503977649067932009887563, 3.56873664642378457138297166577, 4.60351184135207820401471562058, 5.18855266961792966435476949617, 6.38697633231729836443791911502, 7.33687862727726790118417653000, 8.083256344440692245142999818639, 8.771572869926970143173379865497, 9.588814207360503313943459944371

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