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Properties

Label 2-1472-368.45-c0-0-0
Degree $2$
Conductor $1472$
Sign $0.130 - 0.991i$
Analytic cond. $0.734623$
Root an. cond. $0.857101$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 1.36i)3-s + 2.73i·9-s + (−0.366 − 0.366i)13-s − i·23-s − i·25-s + (−2.36 + 2.36i)27-s + (1.36 + 1.36i)29-s − 1.73·31-s − i·39-s − i·41-s + 47-s − 49-s + (1 − i)59-s + (1.36 − 1.36i)69-s − 1.73i·71-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)3-s + 2.73i·9-s + (−0.366 − 0.366i)13-s − i·23-s − i·25-s + (−2.36 + 2.36i)27-s + (1.36 + 1.36i)29-s − 1.73·31-s − i·39-s − i·41-s + 47-s − 49-s + (1 − i)59-s + (1.36 − 1.36i)69-s − 1.73i·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (1425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :0),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.658257056\)
\(L(\frac12)\) \(\approx\) \(1.658257056\)
\(L(1)\) 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 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
29 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
31 \( 1 + 1.73T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1 + i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.825381591052167058075017732320, −9.028987984145465172044491161454, −8.508089998849182011810588266819, −7.79084639925570971761793109776, −6.81040914004167357056579837348, −5.39929587373681317054063679733, −4.70171025933114825966281854797, −3.85154042028488456270290057594, −3.00622692088938132002009658678, −2.14950372428015083013317138813, 1.33310930487965156410094574986, 2.30341736868639320295973446477, 3.20350184328983946501665285744, 4.12724559313902438202457951295, 5.59911715850064338962734609561, 6.56373733740205735566123441484, 7.29049630465938043438290388362, 7.81652026419501100265551175732, 8.631457448103184264052196679000, 9.314748864651358753968177033386

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