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Properties

Label 2-1344-12.11-c1-0-9
Degree $2$
Conductor $1344$
Sign $-0.408 - 0.912i$
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  + (0.707 + 1.58i)3-s − 1.41i·5-s + i·7-s + (−2.00 + 2.23i)9-s − 4.47·11-s + 7.16·13-s + (2.23 − 1.00i)15-s + 7.30i·17-s − 0.837i·19-s + (−1.58 + 0.707i)21-s − 5.65·23-s + 2.99·25-s + (−4.94 − 1.58i)27-s − 1.64i·29-s + 6.32i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.912i)3-s − 0.632i·5-s + 0.377i·7-s + (−0.666 + 0.745i)9-s − 1.34·11-s + 1.98·13-s + (0.577 − 0.258i)15-s + 1.77i·17-s − 0.192i·19-s + (−0.345 + 0.154i)21-s − 1.17·23-s + 0.599·25-s + (−0.952 − 0.304i)27-s − 0.305i·29-s + 1.13i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.408 - 0.912i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.538369002\)
\(L(\frac12)\) \(\approx\) \(1.538369002\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 1.58i)T \)
7 \( 1 - iT \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 + 4.47T + 11T^{2} \)
13 \( 1 - 7.16T + 13T^{2} \)
17 \( 1 - 7.30iT - 17T^{2} \)
19 \( 1 + 0.837iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + 1.64iT - 29T^{2} \)
31 \( 1 - 6.32iT - 31T^{2} \)
37 \( 1 + 4.32T + 37T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 - 8.32iT - 43T^{2} \)
47 \( 1 - 8.94T + 47T^{2} \)
53 \( 1 + 1.18iT - 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + 3.16T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 4.32T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 7.53T + 83T^{2} \)
89 \( 1 - 1.18iT - 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−9.907799298723662775887333673419, −8.847627262995957682818164796038, −8.398949701777038365419308655471, −7.930377523066191997873924239233, −6.25540524147776202840076701970, −5.66983908757288269334781388529, −4.71896576299292718196540871505, −3.86284322982223554414240495363, −2.95312333851202765781656167558, −1.60149588652504937716983562791, 0.59968126700808243635757917506, 2.10586073633968617390580167157, 3.03683486907852500215620963613, 3.90413277278193097054655738253, 5.40905776245077986497016393902, 6.13636956361570870438600064691, 7.10843649495373928036527859815, 7.58598151896343075970226669436, 8.463295913470158964975548058442, 9.147757295928412083082372056427

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