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Properties

Label 2-1344-48.11-c1-0-11
Degree $2$
Conductor $1344$
Sign $0.879 - 0.475i$
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  + (−1.66 − 0.490i)3-s + (2.27 − 2.27i)5-s − 7-s + (2.51 + 1.62i)9-s + (1.28 + 1.28i)11-s + (−2.15 + 2.15i)13-s + (−4.88 + 2.65i)15-s + 6.50i·17-s + (3.42 + 3.42i)19-s + (1.66 + 0.490i)21-s + 5.60i·23-s − 5.30i·25-s + (−3.38 − 3.94i)27-s + (−3.59 − 3.59i)29-s + 0.730i·31-s + ⋯
L(s)  = 1  + (−0.959 − 0.283i)3-s + (1.01 − 1.01i)5-s − 0.377·7-s + (0.839 + 0.543i)9-s + (0.388 + 0.388i)11-s + (−0.597 + 0.597i)13-s + (−1.26 + 0.686i)15-s + 1.57i·17-s + (0.785 + 0.785i)19-s + (0.362 + 0.107i)21-s + 1.16i·23-s − 1.06i·25-s + (−0.651 − 0.758i)27-s + (−0.667 − 0.667i)29-s + 0.131i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.879 - 0.475i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.277922974\)
\(L(\frac12)\) \(\approx\) \(1.277922974\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 0.490i)T \)
7 \( 1 + T \)
good5 \( 1 + (-2.27 + 2.27i)T - 5iT^{2} \)
11 \( 1 + (-1.28 - 1.28i)T + 11iT^{2} \)
13 \( 1 + (2.15 - 2.15i)T - 13iT^{2} \)
17 \( 1 - 6.50iT - 17T^{2} \)
19 \( 1 + (-3.42 - 3.42i)T + 19iT^{2} \)
23 \( 1 - 5.60iT - 23T^{2} \)
29 \( 1 + (3.59 + 3.59i)T + 29iT^{2} \)
31 \( 1 - 0.730iT - 31T^{2} \)
37 \( 1 + (-7.94 - 7.94i)T + 37iT^{2} \)
41 \( 1 + 3.23T + 41T^{2} \)
43 \( 1 + (-8.55 + 8.55i)T - 43iT^{2} \)
47 \( 1 + 7.39T + 47T^{2} \)
53 \( 1 + (0.785 - 0.785i)T - 53iT^{2} \)
59 \( 1 + (-4.42 - 4.42i)T + 59iT^{2} \)
61 \( 1 + (1.23 - 1.23i)T - 61iT^{2} \)
67 \( 1 + (-2.62 - 2.62i)T + 67iT^{2} \)
71 \( 1 + 1.31iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + 9.74iT - 79T^{2} \)
83 \( 1 + (0.603 - 0.603i)T - 83iT^{2} \)
89 \( 1 + 0.657T + 89T^{2} \)
97 \( 1 - 8.66T + 97T^{2} \)
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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

−9.832646847859199750270133082359, −9.080867412704305222059889030116, −7.992878162875198884686964974882, −7.10609124066833204365600393591, −6.09052432123674294152971082371, −5.69569417981128532222530884835, −4.77533280772393779883982150517, −3.85680061845245668196683502393, −1.98732260204027408326660622177, −1.26628547903571409388683547686, 0.65773214471321656222287330979, 2.44643664605665411380718264478, 3.26917694354675835374913430304, 4.67881298649497173571358796235, 5.46917281993976482116602456511, 6.25118746093456893987744552443, 6.88518259356980514677423091365, 7.59179953790075130714079016158, 9.268085106534359390142000231206, 9.562994701561124526048259355844

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