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Properties

Label 2-1472-16.5-c1-0-40
Degree $2$
Conductor $1472$
Sign $-0.972 - 0.233i$
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  + (1.70 − 1.70i)3-s + (−2.61 − 2.61i)5-s + 1.66i·7-s − 2.81i·9-s + (−3.35 − 3.35i)11-s + (1.50 − 1.50i)13-s − 8.91·15-s − 0.812·17-s + (3.81 − 3.81i)19-s + (2.83 + 2.83i)21-s − i·23-s + 8.64i·25-s + (0.307 + 0.307i)27-s + (−7.12 + 7.12i)29-s − 10.7·31-s + ⋯
L(s)  = 1  + (0.984 − 0.984i)3-s + (−1.16 − 1.16i)5-s + 0.628i·7-s − 0.939i·9-s + (−1.01 − 1.01i)11-s + (0.418 − 0.418i)13-s − 2.30·15-s − 0.196·17-s + (0.876 − 0.876i)19-s + (0.618 + 0.618i)21-s − 0.208i·23-s + 1.72i·25-s + (0.0591 + 0.0591i)27-s + (−1.32 + 1.32i)29-s − 1.92·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $-0.972 - 0.233i$
Analytic conductor: \(11.7539\)
Root analytic conductor: \(3.42840\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :1/2),\ -0.972 - 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.056985164\)
\(L(\frac12)\) \(\approx\) \(1.056985164\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + (-1.70 + 1.70i)T - 3iT^{2} \)
5 \( 1 + (2.61 + 2.61i)T + 5iT^{2} \)
7 \( 1 - 1.66iT - 7T^{2} \)
11 \( 1 + (3.35 + 3.35i)T + 11iT^{2} \)
13 \( 1 + (-1.50 + 1.50i)T - 13iT^{2} \)
17 \( 1 + 0.812T + 17T^{2} \)
19 \( 1 + (-3.81 + 3.81i)T - 19iT^{2} \)
29 \( 1 + (7.12 - 7.12i)T - 29iT^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 + (-3.80 - 3.80i)T + 37iT^{2} \)
41 \( 1 - 0.765iT - 41T^{2} \)
43 \( 1 + (3.75 + 3.75i)T + 43iT^{2} \)
47 \( 1 + 2.18T + 47T^{2} \)
53 \( 1 + (1.48 + 1.48i)T + 53iT^{2} \)
59 \( 1 + (9.46 + 9.46i)T + 59iT^{2} \)
61 \( 1 + (-0.442 + 0.442i)T - 61iT^{2} \)
67 \( 1 + (-7.97 + 7.97i)T - 67iT^{2} \)
71 \( 1 + 2.88iT - 71T^{2} \)
73 \( 1 - 7.28iT - 73T^{2} \)
79 \( 1 + 1.36T + 79T^{2} \)
83 \( 1 + (6.09 - 6.09i)T - 83iT^{2} \)
89 \( 1 + 4.98iT - 89T^{2} \)
97 \( 1 + 7.46T + 97T^{2} \)
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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

−8.794157732099754003060855946518, −8.270949548885206774934914665835, −7.71521767945720023507613177080, −7.00893884023223518389403778888, −5.58622615137232815089757324156, −5.01705453403118562511552389832, −3.59322272641916278295060255289, −2.96166038542315889153484866352, −1.63936963922318427194120845881, −0.35943461661528111822623406829, 2.19705680088730852899014877330, 3.26901833931389464762193692457, 3.86160231954417103554387337780, 4.45248527815574227023222262254, 5.77314237660522884851775501797, 7.17225774410390131614827715130, 7.53277798681361663611466804839, 8.170172085805824562404391001484, 9.324936086670422878121236538456, 9.891378788878660018433069377975

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