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Properties

Label 2-1472-184.91-c1-0-22
Degree $2$
Conductor $1472$
Sign $0.642 - 0.766i$
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

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.008414513\)
\(L(\frac12)\) \(\approx\) \(2.008414513\)
\(L(\frac{3}{2})\) 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

−9.584974257252368618449143738909, −8.763376274767522402359894486730, −8.194033116166375418214988423050, −7.11627203995642056641705208982, −6.18929028152082289365058663829, −5.43339691574497780432352859091, −4.87035447052511975760975508401, −3.70049201519211145456352688670, −2.03482280682074879462079665507, −1.60168617510387420859402197888, 0.874630623731627651934910544336, 2.16030924045863488564553249258, 3.05104457174969722041198038688, 4.85045655391259506091741134544, 5.14198397421271581781061528590, 5.89886644954073417584108827230, 6.89018688114799101517958680685, 7.971892456040943150830253217441, 8.514658381325705587139536038679, 9.346326636183088279590470952288

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