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Properties

Label 2-1344-8.5-c3-0-31
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 + 6.24i·5-s − 7·7-s − 9·9-s + 63.4i·11-s + 82.2i·13-s − 18.7·15-s + 75.4·17-s + 125. i·19-s − 21i·21-s + 155.·23-s + 85.9·25-s − 27i·27-s − 56.2i·29-s + 159.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.558i·5-s − 0.377·7-s − 0.333·9-s + 1.73i·11-s + 1.75i·13-s − 0.322·15-s + 1.07·17-s + 1.51i·19-s − 0.218i·21-s + 1.40·23-s + 0.687·25-s − 0.192i·27-s − 0.360i·29-s + 0.922·31-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

\(L(2)\) \(\approx\) \(2.107989097\)
\(L(\frac12)\) \(\approx\) \(2.107989097\)
\(L(\frac{5}{2})\) 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 \( 1 - 6.24iT - 125T^{2} \)
11 \( 1 - 63.4iT - 1.33e3T^{2} \)
13 \( 1 - 82.2iT - 2.19e3T^{2} \)
17 \( 1 - 75.4T + 4.91e3T^{2} \)
19 \( 1 - 125. iT - 6.85e3T^{2} \)
23 \( 1 - 155.T + 1.21e4T^{2} \)
29 \( 1 + 56.2iT - 2.43e4T^{2} \)
31 \( 1 - 159.T + 2.97e4T^{2} \)
37 \( 1 - 197. iT - 5.06e4T^{2} \)
41 \( 1 - 137.T + 6.89e4T^{2} \)
43 \( 1 + 295. iT - 7.95e4T^{2} \)
47 \( 1 + 186.T + 1.03e5T^{2} \)
53 \( 1 + 409. iT - 1.48e5T^{2} \)
59 \( 1 - 311. iT - 2.05e5T^{2} \)
61 \( 1 + 168. iT - 2.26e5T^{2} \)
67 \( 1 - 563. iT - 3.00e5T^{2} \)
71 \( 1 + 282.T + 3.57e5T^{2} \)
73 \( 1 - 250.T + 3.89e5T^{2} \)
79 \( 1 - 948.T + 4.93e5T^{2} \)
83 \( 1 - 1.28e3iT - 5.71e5T^{2} \)
89 \( 1 + 655.T + 7.04e5T^{2} \)
97 \( 1 - 706.T + 9.12e5T^{2} \)
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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

−9.812897803854911082252690119097, −9.024189363820534269372645224793, −7.959114155955594882667278045676, −6.98341677756868747047415395661, −6.55172608522606998288242417809, −5.30850894590168269345177598142, −4.45149746015621902728597287257, −3.66234757483517958143886800467, −2.55780697545724965114212909769, −1.44572128912206807540619617304, 0.65497742522278550006028834814, 0.904322841387833893675568100822, 2.97366446279163473743637154511, 3.14093327168763416081546112903, 4.86151484405745204322760033334, 5.57456980216185993072429575629, 6.29891555973120508479861908719, 7.33314722352920252497554536435, 8.131787389163350396167210417505, 8.747422637020138615408352164954

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