Properties

Degree $2$
Conductor $1344$
Sign $0.386 + 0.922i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (1.5 − 2.59i)5-s + (0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 4·13-s + 3·15-s + (−2 − 3.46i)17-s + (2.5 − 0.866i)21-s + (−4 + 6.92i)23-s + (−2 − 3.46i)25-s − 0.999·27-s + 7·29-s + (−5.5 − 9.52i)31-s + (0.499 − 0.866i)33-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.670 − 1.16i)5-s + (0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.150 − 0.261i)11-s + 1.10·13-s + 0.774·15-s + (−0.485 − 0.840i)17-s + (0.545 − 0.188i)21-s + (−0.834 + 1.44i)23-s + (−0.400 − 0.692i)25-s − 0.192·27-s + 1.29·29-s + (−0.987 − 1.71i)31-s + (0.0870 − 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.386 + 0.922i$
Motivic weight: \(1\)
Character: $\chi_{1344} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.032615435\)
\(L(\frac12)\) \(\approx\) \(2.032615435\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.608586229737677162927271317162, −8.657346263488592570553212705801, −8.093482024827008278016674666494, −7.08683249791056411194129784235, −5.92284214698367475953356895183, −5.22079413497437144735511355551, −4.29059236587322857049971367076, −3.56082050619406454394368842853, −2.02653109127812968674929359939, −0.827964424476160638503126688935, 1.68529339660953502136328108613, 2.52015344641385632819445124923, 3.37357076077600121728009330204, 4.72869598950172916959730275789, 6.09231864265076290503379103225, 6.26865157089882679121800009773, 7.17635913379811346826055339069, 8.462824954838264001694209489614, 8.613086159089839350282324759769, 9.838693851739017473414669223754

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