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Properties

Label 2-1344-7.4-c1-0-14
Degree $2$
Conductor $1344$
Sign $0.605 + 0.795i$
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  + (−0.5 + 0.866i)3-s + (−2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s − 13-s + (1 − 1.73i)17-s + (2.5 + 4.33i)19-s + (2 − 1.73i)21-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s + 0.999·27-s + 8·29-s + (1.5 − 2.59i)31-s + (−0.999 − 1.73i)33-s + (−4.5 − 7.79i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s − 0.277·13-s + (0.242 − 0.420i)17-s + (0.573 + 0.993i)19-s + (0.436 − 0.377i)21-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + 0.192·27-s + 1.48·29-s + (0.269 − 0.466i)31-s + (−0.174 − 0.301i)33-s + (−0.739 − 1.28i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9816928710\)
\(L(\frac12)\) \(\approx\) \(0.9816928710\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.5 + 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−9.848589323676962205290741605351, −8.763120175094757912631920692655, −7.889774898031622798111438929047, −6.90340070201399535111045252086, −6.24018333968719341857591604218, −5.24328481503689659144603232782, −4.35983985897812890404926196284, −3.44422768050030759813274474217, −2.37348506301054889681060003012, −0.48043511712562793718704805138, 1.13109483239319224487879201996, 2.68423668223286424696045265535, 3.40446873788660811878111012004, 4.84587181434337830202536018489, 5.66063025990894928977525669229, 6.47453539411374882413216972764, 7.16633531331130568745747850705, 8.093126505501376770573794505617, 8.920591842914791714198456988879, 9.736911018352367137765343386626

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