Properties

Label 2-1344-7.2-c1-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−2 + 3.46i)5-s + (2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s − 5·13-s − 3.99·15-s + (−1 − 1.73i)17-s + (−0.5 + 0.866i)19-s + (2 + 1.73i)21-s + (−3 + 5.19i)23-s + (−5.49 − 9.52i)25-s − 0.999·27-s + (−1.5 − 2.59i)31-s + (3 − 5.19i)33-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.894 + 1.54i)5-s + (0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s − 1.38·13-s − 1.03·15-s + (−0.242 − 0.420i)17-s + (−0.114 + 0.198i)19-s + (0.436 + 0.377i)21-s + (−0.625 + 1.08i)23-s + (−1.09 − 1.90i)25-s − 0.192·27-s + (−0.269 − 0.466i)31-s + (0.522 − 0.904i)33-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: $\chi_{1344} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
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 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.419382354141753284742995101688, −8.223373703662598902568952040242, −7.73981743416146513036747746327, −7.18554149881087138515494660794, −5.96573570114839498057923853526, −5.01923809027054233052667864290, −3.97631658247478653323370928207, −3.16042618390013366449468201205, −2.35840850751405595125872558675, 0, 1.61916457107724997243687807398, 2.49305908448567367030785356109, 4.27341296974478721417591623125, 4.74595426399781404789886099900, 5.39448727056468430255608451844, 6.94175021694050638782766532680, 7.73430200867319171148577948312, 8.137627364123077207530094041446, 8.888894369879400097993493848140

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