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Properties

Label 2-1344-8.5-c3-0-39
Degree $2$
Conductor $1344$
Sign $0.965 + 0.258i$
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 − 17.4i·5-s + 7·7-s − 9·9-s + 37.3i·11-s − 4.32i·13-s + 52.4·15-s − 0.453·17-s − 58.2i·19-s + 21i·21-s + 181.·23-s − 180.·25-s − 27i·27-s + 255. i·29-s + 30.0·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.56i·5-s + 0.377·7-s − 0.333·9-s + 1.02i·11-s − 0.0922i·13-s + 0.902·15-s − 0.00646·17-s − 0.703i·19-s + 0.218i·21-s + 1.64·23-s − 1.44·25-s − 0.192i·27-s + 1.63i·29-s + 0.174·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.965 + 0.258i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.207169763\)
\(L(\frac12)\) \(\approx\) \(2.207169763\)
\(L(\frac{5}{2})\) 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 \( 1 + 17.4iT - 125T^{2} \)
11 \( 1 - 37.3iT - 1.33e3T^{2} \)
13 \( 1 + 4.32iT - 2.19e3T^{2} \)
17 \( 1 + 0.453T + 4.91e3T^{2} \)
19 \( 1 + 58.2iT - 6.85e3T^{2} \)
23 \( 1 - 181.T + 1.21e4T^{2} \)
29 \( 1 - 255. iT - 2.43e4T^{2} \)
31 \( 1 - 30.0T + 2.97e4T^{2} \)
37 \( 1 + 135. iT - 5.06e4T^{2} \)
41 \( 1 - 30.5T + 6.89e4T^{2} \)
43 \( 1 - 423. iT - 7.95e4T^{2} \)
47 \( 1 - 28.9T + 1.03e5T^{2} \)
53 \( 1 - 117. iT - 1.48e5T^{2} \)
59 \( 1 + 447. iT - 2.05e5T^{2} \)
61 \( 1 + 22.4iT - 2.26e5T^{2} \)
67 \( 1 - 246. iT - 3.00e5T^{2} \)
71 \( 1 - 56.3T + 3.57e5T^{2} \)
73 \( 1 - 268.T + 3.89e5T^{2} \)
79 \( 1 - 964.T + 4.93e5T^{2} \)
83 \( 1 + 1.28e3iT - 5.71e5T^{2} \)
89 \( 1 + 269.T + 7.04e5T^{2} \)
97 \( 1 - 968.T + 9.12e5T^{2} \)
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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

−9.121106527436969411209662070348, −8.679310485140094850563937321882, −7.71239441477687764745773986895, −6.82286718894997615484377423421, −5.50052000103767852951605385752, −4.82379962671233014297440549419, −4.46610888944598079205477292682, −3.12954850534925467100626408998, −1.74351787579786901068288725273, −0.72908375813666360795147149113, 0.789204327896033000172903010080, 2.18476326102974101887184387944, 3.01114348808281359529338405941, 3.85583689863347444389122622080, 5.29824471674198602630168114305, 6.19151771630235485725786770183, 6.77875015614783977486579042012, 7.59943778751865705812768841202, 8.267572702975987085914580708581, 9.233201433865807381448040486312

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