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Properties

Label 2-1344-48.11-c1-0-18
Degree $2$
Conductor $1344$
Sign $0.777 - 0.628i$
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  + (−1.34 + 1.09i)3-s + (−1.07 + 1.07i)5-s − 7-s + (0.620 − 2.93i)9-s + (2.21 + 2.21i)11-s + (3.25 − 3.25i)13-s + (0.273 − 2.61i)15-s − 2.00i·17-s + (−2.74 − 2.74i)19-s + (1.34 − 1.09i)21-s − 3.98i·23-s + 2.69i·25-s + (2.36 + 4.62i)27-s + (3.66 + 3.66i)29-s − 7.29i·31-s + ⋯
L(s)  = 1  + (−0.776 + 0.629i)3-s + (−0.480 + 0.480i)5-s − 0.377·7-s + (0.206 − 0.978i)9-s + (0.666 + 0.666i)11-s + (0.902 − 0.902i)13-s + (0.0705 − 0.675i)15-s − 0.485i·17-s + (−0.630 − 0.630i)19-s + (0.293 − 0.238i)21-s − 0.831i·23-s + 0.538i·25-s + (0.455 + 0.890i)27-s + (0.681 + 0.681i)29-s − 1.31i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.777 - 0.628i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.777 - 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.108065418\)
\(L(\frac12)\) \(\approx\) \(1.108065418\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 1.09i)T \)
7 \( 1 + T \)
good5 \( 1 + (1.07 - 1.07i)T - 5iT^{2} \)
11 \( 1 + (-2.21 - 2.21i)T + 11iT^{2} \)
13 \( 1 + (-3.25 + 3.25i)T - 13iT^{2} \)
17 \( 1 + 2.00iT - 17T^{2} \)
19 \( 1 + (2.74 + 2.74i)T + 19iT^{2} \)
23 \( 1 + 3.98iT - 23T^{2} \)
29 \( 1 + (-3.66 - 3.66i)T + 29iT^{2} \)
31 \( 1 + 7.29iT - 31T^{2} \)
37 \( 1 + (-4.74 - 4.74i)T + 37iT^{2} \)
41 \( 1 - 2.69T + 41T^{2} \)
43 \( 1 + (1.69 - 1.69i)T - 43iT^{2} \)
47 \( 1 - 0.231T + 47T^{2} \)
53 \( 1 + (5.54 - 5.54i)T - 53iT^{2} \)
59 \( 1 + (-10.1 - 10.1i)T + 59iT^{2} \)
61 \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \)
67 \( 1 + (-10.6 - 10.6i)T + 67iT^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 - 5.10iT - 73T^{2} \)
79 \( 1 - 0.216iT - 79T^{2} \)
83 \( 1 + (-3.49 + 3.49i)T - 83iT^{2} \)
89 \( 1 - 1.48T + 89T^{2} \)
97 \( 1 + 5.51T + 97T^{2} \)
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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

−9.820192066894527853162578566374, −9.054951809554224731621002140087, −8.115805412956421998368442958065, −6.96963741211026280527740309620, −6.47929765394355315398216200144, −5.54731587174588878898419861753, −4.50554149672222341195637194577, −3.78651029320628780911033331700, −2.76612307539555256864269062185, −0.825992445395922585276314406474, 0.799199202079873159910554260827, 1.93380238888101124140439718666, 3.61708885954291769403608017795, 4.35435111442947118878377390481, 5.51960898051552447613482292576, 6.34844497689258602258836341306, 6.79454892174673588843009925517, 8.056458035953797761584610175742, 8.494517083598002786915379817716, 9.477641769103821793380878179438

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