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Properties

Label 2-1344-28.3-c1-0-26
Degree $2$
Conductor $1344$
Sign $-0.677 + 0.735i$
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 + (−0.380 + 0.219i)5-s + (2.02 − 1.70i)7-s + (−0.499 − 0.866i)9-s + (−1.83 − 1.05i)11-s − 3.84i·13-s − 0.438i·15-s + (−4.89 − 2.82i)17-s + (1.48 + 2.57i)19-s + (0.464 + 2.60i)21-s + (−4.13 + 2.38i)23-s + (−2.40 + 4.16i)25-s + 0.999·27-s − 7.02·29-s + (−3.71 + 6.43i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.170 + 0.0981i)5-s + (0.764 − 0.644i)7-s + (−0.166 − 0.288i)9-s + (−0.552 − 0.318i)11-s − 1.06i·13-s − 0.113i·15-s + (−1.18 − 0.684i)17-s + (0.341 + 0.591i)19-s + (0.101 + 0.568i)21-s + (−0.861 + 0.497i)23-s + (−0.480 + 0.832i)25-s + 0.192·27-s − 1.30·29-s + (−0.666 + 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.677 + 0.735i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.677 + 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4989318195\)
\(L(\frac12)\) \(\approx\) \(0.4989318195\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 1.70i)T \)
good5 \( 1 + (0.380 - 0.219i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.83 + 1.05i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.84iT - 13T^{2} \)
17 \( 1 + (4.89 + 2.82i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.48 - 2.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.13 - 2.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.02T + 29T^{2} \)
31 \( 1 + (3.71 - 6.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.64 + 4.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.81iT - 41T^{2} \)
43 \( 1 + 4.38iT - 43T^{2} \)
47 \( 1 + (0.844 + 1.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.35 + 9.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.05 - 7.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.35 - 3.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.79 - 3.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.16iT - 71T^{2} \)
73 \( 1 + (8.69 + 5.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.4 + 7.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.49T + 83T^{2} \)
89 \( 1 + (9.02 - 5.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.22iT - 97T^{2} \)
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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

−9.347037199445148740108486096992, −8.506154811826060798091943457543, −7.61600057066581918858274438669, −7.07986326336282007022006613918, −5.60887159618244079472509527520, −5.28461218004257683965359472130, −4.08540545459923934492897625832, −3.35980829258018654199859940335, −1.89007662242200476272781832417, −0.19918339917908269558996081354, 1.76036016625971485532555233889, 2.44354951110861199444382502010, 4.14060401552591275900189937078, 4.80391861304740289876177929604, 5.86882736102748069842648850036, 6.55926814403362612804549878898, 7.58051360073034121382235820827, 8.204914177136978730827888091389, 9.011094552412730937418351519018, 9.816449864860648111595789706804

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