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Properties

Label 2-1344-48.11-c1-0-35
Degree $2$
Conductor $1344$
Sign $0.975 + 0.221i$
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  + (1.65 + 0.505i)3-s + (2.00 − 2.00i)5-s − 7-s + (2.48 + 1.67i)9-s + (2.67 + 2.67i)11-s + (4.14 − 4.14i)13-s + (4.34 − 2.31i)15-s + 3.45i·17-s + (−5.27 − 5.27i)19-s + (−1.65 − 0.505i)21-s − 2.11i·23-s − 3.06i·25-s + (3.27 + 4.03i)27-s + (−0.808 − 0.808i)29-s + 7.56i·31-s + ⋯
L(s)  = 1  + (0.956 + 0.291i)3-s + (0.897 − 0.897i)5-s − 0.377·7-s + (0.829 + 0.558i)9-s + (0.806 + 0.806i)11-s + (1.14 − 1.14i)13-s + (1.12 − 0.596i)15-s + 0.838i·17-s + (−1.21 − 1.21i)19-s + (−0.361 − 0.110i)21-s − 0.441i·23-s − 0.612i·25-s + (0.630 + 0.776i)27-s + (−0.150 − 0.150i)29-s + 1.35i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.975 + 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.873280585\)
\(L(\frac12)\) \(\approx\) \(2.873280585\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 0.505i)T \)
7 \( 1 + T \)
good5 \( 1 + (-2.00 + 2.00i)T - 5iT^{2} \)
11 \( 1 + (-2.67 - 2.67i)T + 11iT^{2} \)
13 \( 1 + (-4.14 + 4.14i)T - 13iT^{2} \)
17 \( 1 - 3.45iT - 17T^{2} \)
19 \( 1 + (5.27 + 5.27i)T + 19iT^{2} \)
23 \( 1 + 2.11iT - 23T^{2} \)
29 \( 1 + (0.808 + 0.808i)T + 29iT^{2} \)
31 \( 1 - 7.56iT - 31T^{2} \)
37 \( 1 + (1.06 + 1.06i)T + 37iT^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + (-5.32 + 5.32i)T - 43iT^{2} \)
47 \( 1 - 6.19T + 47T^{2} \)
53 \( 1 + (-0.414 + 0.414i)T - 53iT^{2} \)
59 \( 1 + (-7.26 - 7.26i)T + 59iT^{2} \)
61 \( 1 + (1.06 - 1.06i)T - 61iT^{2} \)
67 \( 1 + (4.81 + 4.81i)T + 67iT^{2} \)
71 \( 1 + 1.83iT - 71T^{2} \)
73 \( 1 - 0.150iT - 73T^{2} \)
79 \( 1 + 4.73iT - 79T^{2} \)
83 \( 1 + (1.94 - 1.94i)T - 83iT^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−9.360535879329686305274643819380, −8.767403555604221613916512753531, −8.416430662172927004070074099655, −7.12316720857001884557665297516, −6.30884527191084027033579016709, −5.30117747906070087303504184875, −4.37669872571194634986825246217, −3.51239744215240878449383024095, −2.27841035341960252927956312049, −1.29080421042356999069166311896, 1.48758847948154525205536490105, 2.40675173332390706521014258111, 3.48932195695841294544038301865, 4.11178258117834658135193311074, 5.89434007438563161357972653300, 6.41952438835525632315631800187, 7.01231173672621672525093254996, 8.169894075817462553870355729983, 8.906525972857224562106235332233, 9.534401458983818678148911813413

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