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Properties

Label 2-1472-184.91-c1-0-25
Degree $2$
Conductor $1472$
Sign $0.820 - 0.571i$
Analytic cond. $11.7539$
Root an. cond. $3.42840$
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  + 3.22·3-s − 3.21·5-s + 2.55·7-s + 7.39·9-s + 5.44i·11-s + 2.85i·13-s − 10.3·15-s − 6.49i·17-s + 2.41i·19-s + 8.25·21-s + (1.63 + 4.51i)23-s + 5.32·25-s + 14.1·27-s − 4.62i·29-s + 2.88i·31-s + ⋯
L(s)  = 1  + 1.86·3-s − 1.43·5-s + 0.967·7-s + 2.46·9-s + 1.64i·11-s + 0.791i·13-s − 2.67·15-s − 1.57i·17-s + 0.554i·19-s + 1.80·21-s + (0.339 + 0.940i)23-s + 1.06·25-s + 2.72·27-s − 0.859i·29-s + 0.518i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $0.820 - 0.571i$
Analytic conductor: \(11.7539\)
Root analytic conductor: \(3.42840\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :1/2),\ 0.820 - 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.904018056\)
\(L(\frac12)\) \(\approx\) \(2.904018056\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (-1.63 - 4.51i)T \)
good3 \( 1 - 3.22T + 3T^{2} \)
5 \( 1 + 3.21T + 5T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 - 5.44iT - 11T^{2} \)
13 \( 1 - 2.85iT - 13T^{2} \)
17 \( 1 + 6.49iT - 17T^{2} \)
19 \( 1 - 2.41iT - 19T^{2} \)
29 \( 1 + 4.62iT - 29T^{2} \)
31 \( 1 - 2.88iT - 31T^{2} \)
37 \( 1 - 5.83T + 37T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 - 5.84iT - 43T^{2} \)
47 \( 1 + 7.75iT - 47T^{2} \)
53 \( 1 + 6.88T + 53T^{2} \)
59 \( 1 - 9.91T + 59T^{2} \)
61 \( 1 - 9.06T + 61T^{2} \)
67 \( 1 + 1.63iT - 67T^{2} \)
71 \( 1 - 3.58iT - 71T^{2} \)
73 \( 1 + 7.69T + 73T^{2} \)
79 \( 1 + 0.0691T + 79T^{2} \)
83 \( 1 - 2.38iT - 83T^{2} \)
89 \( 1 - 0.0691iT - 89T^{2} \)
97 \( 1 + 3.16iT - 97T^{2} \)
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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

−9.548602436212764814685649985638, −8.581111013564272985120779719251, −7.999076022917592096786630524921, −7.31915369924679482184147347888, −7.06616967496003508481598123902, −4.84886733141977467115241188013, −4.37582453802911204904969081481, −3.58790189115344258501883912854, −2.51432669175331001435687764353, −1.56282554538000862552719425377, 1.05516042077596386802876533617, 2.52023992739142033124390557674, 3.45116047165754704492118359972, 3.93537912952166647092021751809, 4.91820798911389672053698840468, 6.36570501875812161339971074849, 7.60420355151267564667948084009, 7.934651641509621515464195198605, 8.630246033299179908621974947290, 8.794785986887414306118474135374

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