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Properties

Label 2-1152-48.11-c1-0-8
Degree $2$
Conductor $1152$
Sign $0.965 - 0.261i$
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  + (0.236 − 0.236i)5-s + 3.27·7-s + (2.58 + 2.58i)11-s + (1.70 − 1.70i)13-s + 7.05i·17-s + (−3.04 − 3.04i)19-s − 1.47i·23-s + 4.88i·25-s + (−2.98 − 2.98i)29-s − 8.02i·31-s + (0.774 − 0.774i)35-s + (7.93 + 7.93i)37-s − 2.22·41-s + (4.61 − 4.61i)43-s + 7.13·47-s + ⋯
L(s)  = 1  + (0.105 − 0.105i)5-s + 1.23·7-s + (0.778 + 0.778i)11-s + (0.473 − 0.473i)13-s + 1.71i·17-s + (−0.697 − 0.697i)19-s − 0.307i·23-s + 0.977i·25-s + (−0.554 − 0.554i)29-s − 1.44i·31-s + (0.130 − 0.130i)35-s + (1.30 + 1.30i)37-s − 0.346·41-s + (0.703 − 0.703i)43-s + 1.04·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.965 - 0.261i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.965 - 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.016571539\)
\(L(\frac12)\) \(\approx\) \(2.016571539\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.236 + 0.236i)T - 5iT^{2} \)
7 \( 1 - 3.27T + 7T^{2} \)
11 \( 1 + (-2.58 - 2.58i)T + 11iT^{2} \)
13 \( 1 + (-1.70 + 1.70i)T - 13iT^{2} \)
17 \( 1 - 7.05iT - 17T^{2} \)
19 \( 1 + (3.04 + 3.04i)T + 19iT^{2} \)
23 \( 1 + 1.47iT - 23T^{2} \)
29 \( 1 + (2.98 + 2.98i)T + 29iT^{2} \)
31 \( 1 + 8.02iT - 31T^{2} \)
37 \( 1 + (-7.93 - 7.93i)T + 37iT^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 + (-4.61 + 4.61i)T - 43iT^{2} \)
47 \( 1 - 7.13T + 47T^{2} \)
53 \( 1 + (5.81 - 5.81i)T - 53iT^{2} \)
59 \( 1 + (-7.46 - 7.46i)T + 59iT^{2} \)
61 \( 1 + (-4.04 + 4.04i)T - 61iT^{2} \)
67 \( 1 + (-2.90 - 2.90i)T + 67iT^{2} \)
71 \( 1 + 1.02iT - 71T^{2} \)
73 \( 1 + 4.08iT - 73T^{2} \)
79 \( 1 + 5.36iT - 79T^{2} \)
83 \( 1 + (3.93 - 3.93i)T - 83iT^{2} \)
89 \( 1 - 2.35T + 89T^{2} \)
97 \( 1 - 9.97T + 97T^{2} \)
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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

−9.817557553337347621121177340881, −8.924002712145981606115446132849, −8.197751940208302308963981567480, −7.51373296274007334727889993508, −6.40325675243418332027298913607, −5.63621002916367838986380167384, −4.49536578618771013566513219454, −3.92663579149240372544639150684, −2.27368812997783390818282218400, −1.31544981835083169536754437598, 1.09413107409421158328896617158, 2.30673493799768364240651696865, 3.64472299832049712121340993696, 4.57755376327428790845294466787, 5.48231662657811140757879171529, 6.42485807436007899771674169023, 7.30254396402532000732385783895, 8.229403374505339691878156025281, 8.873837757273089695396055278689, 9.649812963590591762602233951801

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