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Properties

Label 2-1344-8.5-c3-0-62
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 − 11.7i·5-s + 7·7-s − 9·9-s − 72.4i·11-s − 50.7i·13-s + 35.2·15-s + 60.4·17-s − 33.8i·19-s + 21i·21-s + 116.·23-s − 13.1·25-s − 27i·27-s − 13.1i·29-s − 250.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.05i·5-s + 0.377·7-s − 0.333·9-s − 1.98i·11-s − 1.08i·13-s + 0.607·15-s + 0.862·17-s − 0.408i·19-s + 0.218i·21-s + 1.05·23-s − 0.105·25-s − 0.192i·27-s − 0.0844i·29-s − 1.44·31-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

\(L(2)\) \(\approx\) \(1.687537206\)
\(L(\frac12)\) \(\approx\) \(1.687537206\)
\(L(\frac{5}{2})\) 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 \( 1 + 11.7iT - 125T^{2} \)
11 \( 1 + 72.4iT - 1.33e3T^{2} \)
13 \( 1 + 50.7iT - 2.19e3T^{2} \)
17 \( 1 - 60.4T + 4.91e3T^{2} \)
19 \( 1 + 33.8iT - 6.85e3T^{2} \)
23 \( 1 - 116.T + 1.21e4T^{2} \)
29 \( 1 + 13.1iT - 2.43e4T^{2} \)
31 \( 1 + 250.T + 2.97e4T^{2} \)
37 \( 1 - 92.7iT - 5.06e4T^{2} \)
41 \( 1 + 69.1T + 6.89e4T^{2} \)
43 \( 1 + 69.6iT - 7.95e4T^{2} \)
47 \( 1 + 346.T + 1.03e5T^{2} \)
53 \( 1 - 585. iT - 1.48e5T^{2} \)
59 \( 1 + 66.1iT - 2.05e5T^{2} \)
61 \( 1 + 492. iT - 2.26e5T^{2} \)
67 \( 1 + 543. iT - 3.00e5T^{2} \)
71 \( 1 + 365.T + 3.57e5T^{2} \)
73 \( 1 - 374.T + 3.89e5T^{2} \)
79 \( 1 + 670.T + 4.93e5T^{2} \)
83 \( 1 - 595. iT - 5.71e5T^{2} \)
89 \( 1 - 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + 218.T + 9.12e5T^{2} \)
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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

−8.832999577482346417785807433151, −8.362599375544047976033886118499, −7.58335556545201370265986916267, −6.16719014263078314467245894623, −5.36734108497429977835857882822, −4.94477461824578381953513203218, −3.62502650951338126484028111795, −2.98036916461063941588924319760, −1.20911046786029990205897541867, −0.40860185669227677756603194008, 1.51376594263855398858736005683, 2.21038186443268589270622727610, 3.35969749035689206787001635257, 4.46465949250789413520861282403, 5.39252812304442911460188923447, 6.57235464867312573158381614944, 7.17328298725753535607104001720, 7.54005221899838013245685373643, 8.734616664332026526785982283982, 9.639653033916567174488846561233

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