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Properties

Label 2-1344-112.27-c1-0-23
Degree $2$
Conductor $1344$
Sign $-0.00903 + 0.999i$
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  + (0.707 − 0.707i)3-s + (−1.45 + 1.45i)5-s + (2.11 − 1.59i)7-s − 1.00i·9-s + (2.03 − 2.03i)11-s + (−4.25 − 4.25i)13-s + 2.05i·15-s + 2.62i·17-s + (2.19 − 2.19i)19-s + (0.369 − 2.61i)21-s − 4.12·23-s + 0.761i·25-s + (−0.707 − 0.707i)27-s + (5.04 − 5.04i)29-s − 1.60·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.651 + 0.651i)5-s + (0.798 − 0.601i)7-s − 0.333i·9-s + (0.612 − 0.612i)11-s + (−1.18 − 1.18i)13-s + 0.531i·15-s + 0.635i·17-s + (0.504 − 0.504i)19-s + (0.0806 − 0.571i)21-s − 0.860·23-s + 0.152i·25-s + (−0.136 − 0.136i)27-s + (0.936 − 0.936i)29-s − 0.288·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00903 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00903 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.00903 + 0.999i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.00903 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.562780322\)
\(L(\frac12)\) \(\approx\) \(1.562780322\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (-2.11 + 1.59i)T \)
good5 \( 1 + (1.45 - 1.45i)T - 5iT^{2} \)
11 \( 1 + (-2.03 + 2.03i)T - 11iT^{2} \)
13 \( 1 + (4.25 + 4.25i)T + 13iT^{2} \)
17 \( 1 - 2.62iT - 17T^{2} \)
19 \( 1 + (-2.19 + 2.19i)T - 19iT^{2} \)
23 \( 1 + 4.12T + 23T^{2} \)
29 \( 1 + (-5.04 + 5.04i)T - 29iT^{2} \)
31 \( 1 + 1.60T + 31T^{2} \)
37 \( 1 + (6.27 + 6.27i)T + 37iT^{2} \)
41 \( 1 + 2.45T + 41T^{2} \)
43 \( 1 + (-5.72 + 5.72i)T - 43iT^{2} \)
47 \( 1 - 6.44T + 47T^{2} \)
53 \( 1 + (-5.91 - 5.91i)T + 53iT^{2} \)
59 \( 1 + (5.64 + 5.64i)T + 59iT^{2} \)
61 \( 1 + (9.13 + 9.13i)T + 61iT^{2} \)
67 \( 1 + (-4.56 - 4.56i)T + 67iT^{2} \)
71 \( 1 - 1.11T + 71T^{2} \)
73 \( 1 - 4.88T + 73T^{2} \)
79 \( 1 + 12.6iT - 79T^{2} \)
83 \( 1 + (-7.46 + 7.46i)T - 83iT^{2} \)
89 \( 1 - 8.42T + 89T^{2} \)
97 \( 1 + 7.55iT - 97T^{2} \)
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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

−9.327808840406403288449337483775, −8.387336639832798004492066776277, −7.62639038936541592021667969758, −7.32309733323177486515143724542, −6.22596065824663730221707264608, −5.18470353712631107318114161007, −4.04410871752125438659196881627, −3.28542226881853068639199529348, −2.15332419622095448521658483854, −0.62684335450447221192882571762, 1.54761224740051032810498436074, 2.62823325324935409295639443118, 4.03756188266016438489829122930, 4.64535759051737054434135155858, 5.30259886715294478058789476999, 6.69829720512557637656624289351, 7.54282414655587462098233904626, 8.283977349917568517966573882136, 9.045924315243776709699623982420, 9.579357432944310166204802251153

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