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Properties

Label 2-1344-48.11-c1-0-1
Degree $2$
Conductor $1344$
Sign $-0.771 - 0.636i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
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  + (−1.73 + 0.0404i)3-s + (0.0573 − 0.0573i)5-s + 7-s + (2.99 − 0.139i)9-s + (−1.96 − 1.96i)11-s + (−4.48 + 4.48i)13-s + (−0.0969 + 0.101i)15-s − 4.71i·17-s + (2.60 + 2.60i)19-s + (−1.73 + 0.0404i)21-s − 2.30i·23-s + 4.99i·25-s + (−5.18 + 0.363i)27-s + (3.24 + 3.24i)29-s − 1.26i·31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0233i)3-s + (0.0256 − 0.0256i)5-s + 0.377·7-s + (0.998 − 0.0466i)9-s + (−0.592 − 0.592i)11-s + (−1.24 + 1.24i)13-s + (−0.0250 + 0.0262i)15-s − 1.14i·17-s + (0.598 + 0.598i)19-s + (−0.377 + 0.00881i)21-s − 0.479i·23-s + 0.998i·25-s + (−0.997 + 0.0699i)27-s + (0.601 + 0.601i)29-s − 0.227i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.771 - 0.636i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3790240689\)
\(L(\frac12)\) \(\approx\) \(0.3790240689\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.73 - 0.0404i)T \)
7 \( 1 - T \)
good5 \( 1 + (-0.0573 + 0.0573i)T - 5iT^{2} \)
11 \( 1 + (1.96 + 1.96i)T + 11iT^{2} \)
13 \( 1 + (4.48 - 4.48i)T - 13iT^{2} \)
17 \( 1 + 4.71iT - 17T^{2} \)
19 \( 1 + (-2.60 - 2.60i)T + 19iT^{2} \)
23 \( 1 + 2.30iT - 23T^{2} \)
29 \( 1 + (-3.24 - 3.24i)T + 29iT^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + (2.13 + 2.13i)T + 37iT^{2} \)
41 \( 1 + 6.69T + 41T^{2} \)
43 \( 1 + (1.86 - 1.86i)T - 43iT^{2} \)
47 \( 1 + 5.14T + 47T^{2} \)
53 \( 1 + (-1.91 + 1.91i)T - 53iT^{2} \)
59 \( 1 + (-3.22 - 3.22i)T + 59iT^{2} \)
61 \( 1 + (7.78 - 7.78i)T - 61iT^{2} \)
67 \( 1 + (8.73 + 8.73i)T + 67iT^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 8.63iT - 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 + (7.95 - 7.95i)T - 83iT^{2} \)
89 \( 1 + 18.1T + 89T^{2} \)
97 \( 1 - 4.55T + 97T^{2} \)
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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

−9.928848960193105348646853129897, −9.343988189434639242764175530544, −8.244722031507127251688455495968, −7.20894651603424047544457587908, −6.82184110077201183446931882799, −5.52688272060879041272792080861, −5.07346430734185247691892606616, −4.17619426372843713016978058411, −2.77905681124501584326403850024, −1.42319548588645467436513300650, 0.18026340819511904094846113092, 1.75763874153660888627130082590, 3.04406704282034248442673790211, 4.49890913862353386312927191806, 5.06570675314225078791692467143, 5.85262324857823965897542220750, 6.83310480802782390738159804716, 7.63561004267651830086821438748, 8.257953430706069679637523883454, 9.600367635737075006520215497246

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