Properties

Label 2-1344-48.11-c1-0-0
Degree $2$
Conductor $1344$
Sign $-0.978 + 0.206i$
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  + (−1.26 − 1.18i)3-s + (−2.84 + 2.84i)5-s + 7-s + (0.196 + 2.99i)9-s + (3.31 + 3.31i)11-s + (−2.31 + 2.31i)13-s + (6.95 − 0.228i)15-s + 0.814i·17-s + (−4.39 − 4.39i)19-s + (−1.26 − 1.18i)21-s − 3.30i·23-s − 11.1i·25-s + (3.29 − 4.01i)27-s + (3.25 + 3.25i)29-s + 3.89i·31-s + ⋯
L(s)  = 1  + (−0.729 − 0.683i)3-s + (−1.27 + 1.27i)5-s + 0.377·7-s + (0.0656 + 0.997i)9-s + (1.00 + 1.00i)11-s + (−0.641 + 0.641i)13-s + (1.79 − 0.0590i)15-s + 0.197i·17-s + (−1.00 − 1.00i)19-s + (−0.275 − 0.258i)21-s − 0.688i·23-s − 2.23i·25-s + (0.634 − 0.773i)27-s + (0.604 + 0.604i)29-s + 0.699i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.978 + 0.206i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.978 + 0.206i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1504924949\)
\(L(\frac12)\) \(\approx\) \(0.1504924949\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 + 1.18i)T \)
7 \( 1 - T \)
good5 \( 1 + (2.84 - 2.84i)T - 5iT^{2} \)
11 \( 1 + (-3.31 - 3.31i)T + 11iT^{2} \)
13 \( 1 + (2.31 - 2.31i)T - 13iT^{2} \)
17 \( 1 - 0.814iT - 17T^{2} \)
19 \( 1 + (4.39 + 4.39i)T + 19iT^{2} \)
23 \( 1 + 3.30iT - 23T^{2} \)
29 \( 1 + (-3.25 - 3.25i)T + 29iT^{2} \)
31 \( 1 - 3.89iT - 31T^{2} \)
37 \( 1 + (3.65 + 3.65i)T + 37iT^{2} \)
41 \( 1 - 3.20T + 41T^{2} \)
43 \( 1 + (1.63 - 1.63i)T - 43iT^{2} \)
47 \( 1 + 6.70T + 47T^{2} \)
53 \( 1 + (7.69 - 7.69i)T - 53iT^{2} \)
59 \( 1 + (7.31 + 7.31i)T + 59iT^{2} \)
61 \( 1 + (5.54 - 5.54i)T - 61iT^{2} \)
67 \( 1 + (-0.0553 - 0.0553i)T + 67iT^{2} \)
71 \( 1 - 6.50iT - 71T^{2} \)
73 \( 1 + 6.05iT - 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + (-4.25 + 4.25i)T - 83iT^{2} \)
89 \( 1 - 2.31T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−10.43618684677857637066241350517, −9.172928560160446148823857025917, −8.160966380608497407130629050814, −7.34435456917625265725774014739, −6.80870665470712128117899094536, −6.38802420204299826638940302578, −4.72979384590023366469201964641, −4.29982139592181419717532690726, −2.90162779085077791944031221557, −1.78482355863048434603605108486, 0.07653920922882506659891236456, 1.22538905339001002948656816956, 3.42546421876047479350155499580, 4.11428495499231333598306094487, 4.83939110400288487552736694431, 5.63839164684215262962970433939, 6.56749329496085113214672234762, 7.86078760984421171558330951987, 8.337363547026582900945908496755, 9.160679845883459505010737957055

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