Properties

Label 2-1344-48.35-c1-0-23
Degree $2$
Conductor $1344$
Sign $0.975 + 0.220i$
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  + (−1.41 − i)3-s + (2.41 + 2.41i)5-s + 7-s + (1.00 + 2.82i)9-s + (1.82 − 1.82i)11-s + (−1.58 − 1.58i)13-s + (−1 − 5.82i)15-s − 6.82i·17-s + (2.41 − 2.41i)19-s + (−1.41 − i)21-s + 3.65i·23-s + 6.65i·25-s + (1.41 − 5.00i)27-s + (3 − 3i)29-s + 10.4i·31-s + ⋯
L(s)  = 1  + (−0.816 − 0.577i)3-s + (1.07 + 1.07i)5-s + 0.377·7-s + (0.333 + 0.942i)9-s + (0.551 − 0.551i)11-s + (−0.439 − 0.439i)13-s + (−0.258 − 1.50i)15-s − 1.65i·17-s + (0.553 − 0.553i)19-s + (−0.308 − 0.218i)21-s + 0.762i·23-s + 1.33i·25-s + (0.272 − 0.962i)27-s + (0.557 − 0.557i)29-s + 1.88i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.680012956\)
\(L(\frac12)\) \(\approx\) \(1.680012956\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + i)T \)
7 \( 1 - T \)
good5 \( 1 + (-2.41 - 2.41i)T + 5iT^{2} \)
11 \( 1 + (-1.82 + 1.82i)T - 11iT^{2} \)
13 \( 1 + (1.58 + 1.58i)T + 13iT^{2} \)
17 \( 1 + 6.82iT - 17T^{2} \)
19 \( 1 + (-2.41 + 2.41i)T - 19iT^{2} \)
23 \( 1 - 3.65iT - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (3.82 - 3.82i)T - 37iT^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + (1.82 + 1.82i)T + 43iT^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 + (-0.171 - 0.171i)T + 53iT^{2} \)
59 \( 1 + (-4.07 + 4.07i)T - 59iT^{2} \)
61 \( 1 + (0.414 + 0.414i)T + 61iT^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (10.8 + 10.8i)T + 83iT^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.775431799804564253709374446117, −8.886707899886316837341324790307, −7.59773924285304624014419332543, −7.03375832885686614286152040934, −6.34872863457393481072341001023, −5.48545835291658278167263262439, −4.87246908692319516689750274514, −3.15897096146462715546894523367, −2.31050775647320779079563945715, −0.999885880031456449031433174620, 1.10244403189993113319304630940, 2.10912983494208314494426101979, 4.04067543379777126071146160096, 4.52199898647944002307840861968, 5.59348558860047439403756038799, 5.97588474132152370419127524711, 7.00413485143433552534941608471, 8.225047024725725113722899525621, 9.086170645893138695341390865773, 9.651195316338329585832305235775

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