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Properties

Label 2-1472-184.91-c1-0-5
Degree $2$
Conductor $1472$
Sign $-0.766 - 0.642i$
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  + 0.678·3-s + 2.56·5-s − 4.63·7-s − 2.53·9-s + 0.0824i·11-s + 5.74i·13-s + 1.73·15-s + 2.55i·17-s − 4.26i·19-s − 3.14·21-s + (−2.75 − 3.92i)23-s + 1.56·25-s − 3.76·27-s + 4.77i·29-s + 6.46i·31-s + ⋯
L(s)  = 1  + 0.391·3-s + 1.14·5-s − 1.75·7-s − 0.846·9-s + 0.0248i·11-s + 1.59i·13-s + 0.449·15-s + 0.619i·17-s − 0.978i·19-s − 0.686·21-s + (−0.574 − 0.818i)23-s + 0.312·25-s − 0.723·27-s + 0.886i·29-s + 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $-0.766 - 0.642i$
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: \((2,\ 1472,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7391463743\)
\(L(\frac12)\) \(\approx\) \(0.7391463743\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.75 + 3.92i)T \)
good3 \( 1 - 0.678T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 + 4.63T + 7T^{2} \)
11 \( 1 - 0.0824iT - 11T^{2} \)
13 \( 1 - 5.74iT - 13T^{2} \)
17 \( 1 - 2.55iT - 17T^{2} \)
19 \( 1 + 4.26iT - 19T^{2} \)
29 \( 1 - 4.77iT - 29T^{2} \)
31 \( 1 - 6.46iT - 31T^{2} \)
37 \( 1 + 7.41T + 37T^{2} \)
41 \( 1 + 4.66T + 41T^{2} \)
43 \( 1 + 0.420iT - 43T^{2} \)
47 \( 1 - 9.59iT - 47T^{2} \)
53 \( 1 + 9.22T + 53T^{2} \)
59 \( 1 + 2.10T + 59T^{2} \)
61 \( 1 - 6.94T + 61T^{2} \)
67 \( 1 - 0.188iT - 67T^{2} \)
71 \( 1 - 3.17iT - 71T^{2} \)
73 \( 1 - 4.82T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + 7.33iT - 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + 3.42iT - 97T^{2} \)
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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

−9.711122859595870934709739844202, −8.985886235217320488576127711342, −8.679528215131089575224524148586, −7.11389303662125459460946289101, −6.45998709239387935021916973167, −6.00331163913035337624458992670, −4.85762430601031309999171293757, −3.59835163660469423242448825132, −2.78949909021739262601842608451, −1.83185072869029956566155298436, 0.24745130684828136338592466275, 2.13233089006903070858485248365, 3.06254399255493850691436067211, 3.65495193960046644530817197973, 5.49431352517080624762907279021, 5.77619689054682775394670022739, 6.55926323042297021250589849296, 7.66561119233227561144891168170, 8.463572612862026038454651215714, 9.446537136909189897509405966108

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