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Properties

Label 2-1344-28.27-c3-0-70
Degree $2$
Conductor $1344$
Sign $0.581 + 0.813i$
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  + 3·3-s − 15.7i·5-s + (15.0 − 10.7i)7-s + 9·9-s + 63.5i·11-s − 34.9i·13-s − 47.2i·15-s − 39.0i·17-s + 139.·19-s + (45.1 − 32.3i)21-s + 123. i·23-s − 123.·25-s + 27·27-s + 131.·29-s + 225.·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.40i·5-s + (0.813 − 0.581i)7-s + 0.333·9-s + 1.74i·11-s − 0.744i·13-s − 0.813i·15-s − 0.557i·17-s + 1.68·19-s + (0.469 − 0.335i)21-s + 1.11i·23-s − 0.986·25-s + 0.192·27-s + 0.842·29-s + 1.30·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.581 + 0.813i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.409071581\)
\(L(\frac12)\) \(\approx\) \(3.409071581\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 + (-15.0 + 10.7i)T \)
good5 \( 1 + 15.7iT - 125T^{2} \)
11 \( 1 - 63.5iT - 1.33e3T^{2} \)
13 \( 1 + 34.9iT - 2.19e3T^{2} \)
17 \( 1 + 39.0iT - 4.91e3T^{2} \)
19 \( 1 - 139.T + 6.85e3T^{2} \)
23 \( 1 - 123. iT - 1.21e4T^{2} \)
29 \( 1 - 131.T + 2.43e4T^{2} \)
31 \( 1 - 225.T + 2.97e4T^{2} \)
37 \( 1 - 148.T + 5.06e4T^{2} \)
41 \( 1 + 174. iT - 6.89e4T^{2} \)
43 \( 1 - 451. iT - 7.95e4T^{2} \)
47 \( 1 + 146.T + 1.03e5T^{2} \)
53 \( 1 - 278.T + 1.48e5T^{2} \)
59 \( 1 + 26.4T + 2.05e5T^{2} \)
61 \( 1 - 69.5iT - 2.26e5T^{2} \)
67 \( 1 - 288. iT - 3.00e5T^{2} \)
71 \( 1 - 362. iT - 3.57e5T^{2} \)
73 \( 1 + 719. iT - 3.89e5T^{2} \)
79 \( 1 + 972. iT - 4.93e5T^{2} \)
83 \( 1 - 494.T + 5.71e5T^{2} \)
89 \( 1 - 88.7iT - 7.04e5T^{2} \)
97 \( 1 + 372. iT - 9.12e5T^{2} \)
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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

−9.279783426517124107749195193246, −8.085129043527915841762245221316, −7.78028823540155214691033609883, −6.96559740870294847350157697431, −5.40300933354864355005379835344, −4.82182731540456264020504569785, −4.20671842848920600724563134703, −2.87414653541952270839356574968, −1.54969276189020269472760524845, −0.910118806612918498579414129673, 1.06390084122832556453681714901, 2.47572086337388355375985733483, 3.02947912201529332928866320420, 4.00072981496953977086848670300, 5.24740219371412436469295122776, 6.22498107601914962113463901395, 6.84271119034560429976154031539, 7.955343357258040928337371180094, 8.400671072166771872885200034396, 9.271401422366602335954485219299

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