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Properties

Label 2-1344-28.3-c1-0-5
Degree $2$
Conductor $1344$
Sign $-0.311 - 0.950i$
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 + (−1.5 + 0.866i)5-s + (−0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (1.5 + 0.866i)11-s − 1.73i·15-s + (3 + 1.73i)17-s + (1 + 1.73i)19-s + (2.5 + 0.866i)21-s + (−1 + 1.73i)25-s + 0.999·27-s − 9·29-s + (2.5 − 4.33i)31-s + (−1.5 + 0.866i)33-s + (3 + 3.46i)35-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.670 + 0.387i)5-s + (−0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (0.452 + 0.261i)11-s − 0.447i·15-s + (0.727 + 0.420i)17-s + (0.229 + 0.397i)19-s + (0.545 + 0.188i)21-s + (−0.200 + 0.346i)25-s + 0.192·27-s − 1.67·29-s + (0.449 − 0.777i)31-s + (−0.261 + 0.150i)33-s + (0.507 + 0.585i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.311 - 0.950i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9306501406\)
\(L(\frac12)\) \(\approx\) \(0.9306501406\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-3 - 1.73i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12 + 6.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (3 - 1.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−9.837784220773302797077001387848, −9.336637123082073213891484690161, −7.944140995474845982423108013042, −7.58247777544775078486011408921, −6.56038747061640353545031026253, −5.76802753435836959155136317734, −4.52618933989122503672964356492, −3.89062182515766854070625239209, −3.09362576197782670383268073944, −1.26347717082733020909776781681, 0.44094050816918797106739237742, 1.95381724152648213569233911448, 3.16168007392355848246001559615, 4.21359835520414926642259265583, 5.39653681122788288615210998714, 5.88940756302686263316643812330, 7.04089998824801707265326317631, 7.67116182839822955782882178504, 8.659877364889133030308287191103, 9.122798911171030368346284100504

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