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Properties

Label 2-1344-8.5-c3-0-12
Degree $2$
Conductor $1344$
Sign $-0.707 - 0.707i$
Analytic cond. $79.2985$
Root an. cond. $8.90497$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 7.03i·5-s − 7·7-s − 9·9-s − 46.7i·11-s + 8.90i·13-s − 21.1·15-s + 58.0·17-s + 135. i·19-s − 21i·21-s + 86.0·23-s + 75.5·25-s − 27i·27-s − 300. i·29-s − 183.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.629i·5-s − 0.377·7-s − 0.333·9-s − 1.28i·11-s + 0.190i·13-s − 0.363·15-s + 0.828·17-s + 1.63i·19-s − 0.218i·21-s + 0.779·23-s + 0.604·25-s − 0.192i·27-s − 1.92i·29-s − 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.432110760\)
\(L(\frac12)\) \(\approx\) \(1.432110760\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
7 \( 1 + 7T \)
good5 \( 1 - 7.03iT - 125T^{2} \)
11 \( 1 + 46.7iT - 1.33e3T^{2} \)
13 \( 1 - 8.90iT - 2.19e3T^{2} \)
17 \( 1 - 58.0T + 4.91e3T^{2} \)
19 \( 1 - 135. iT - 6.85e3T^{2} \)
23 \( 1 - 86.0T + 1.21e4T^{2} \)
29 \( 1 + 300. iT - 2.43e4T^{2} \)
31 \( 1 + 183.T + 2.97e4T^{2} \)
37 \( 1 - 418. iT - 5.06e4T^{2} \)
41 \( 1 + 106.T + 6.89e4T^{2} \)
43 \( 1 + 162. iT - 7.95e4T^{2} \)
47 \( 1 - 377.T + 1.03e5T^{2} \)
53 \( 1 - 709. iT - 1.48e5T^{2} \)
59 \( 1 - 753. iT - 2.05e5T^{2} \)
61 \( 1 - 667. iT - 2.26e5T^{2} \)
67 \( 1 + 447. iT - 3.00e5T^{2} \)
71 \( 1 + 384.T + 3.57e5T^{2} \)
73 \( 1 - 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + 670.T + 4.93e5T^{2} \)
83 \( 1 + 447. iT - 5.71e5T^{2} \)
89 \( 1 + 70.5T + 7.04e5T^{2} \)
97 \( 1 + 1.31e3T + 9.12e5T^{2} \)
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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

−9.653671046257316414881153284599, −8.760468258269668469478532340457, −8.017282911598744768671541394636, −7.09463000417011460334000375103, −5.99984624180503728499324509959, −5.64583710210800448626604708328, −4.25915620850632322529306563477, −3.42000583547222309076211713570, −2.73511040522965910878549291130, −1.10738205475311295189566331004, 0.36747743491072128712573803816, 1.45344349939748392176494384281, 2.56733922187885275612649820514, 3.65914629664919601728201158161, 4.97325775107496266699092179910, 5.34811974812190473903970112294, 6.88851145734635512290173668795, 7.03184930297746921295988981198, 8.099402597951709386396119211373, 9.112611841393813524464348366212

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