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Properties

Label 2-1344-7.4-c1-0-12
Degree $2$
Conductor $1344$
Sign $0.999 - 0.0226i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
Motivic weight $1$
Arithmetic yes
Rational no
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 + (0.227 + 0.393i)5-s + (2.16 − 1.51i)7-s + (−0.499 − 0.866i)9-s + (−2.89 + 5.01i)11-s + 5.88·13-s + 0.454·15-s + (−1.45 + 2.51i)17-s + (2.94 + 5.09i)19-s + (−0.227 − 2.63i)21-s + (1.45 + 2.51i)23-s + (2.39 − 4.15i)25-s − 0.999·27-s − 3.54·29-s + (2.16 − 3.75i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.101 + 0.176i)5-s + (0.819 − 0.572i)7-s + (−0.166 − 0.288i)9-s + (−0.873 + 1.51i)11-s + 1.63·13-s + 0.117·15-s + (−0.352 + 0.611i)17-s + (0.674 + 1.16i)19-s + (−0.0496 − 0.575i)21-s + (0.303 + 0.525i)23-s + (0.479 − 0.830i)25-s − 0.192·27-s − 0.658·29-s + (0.389 − 0.674i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.999 - 0.0226i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.999 - 0.0226i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.111646622\)
\(L(\frac12)\) \(\approx\) \(2.111646622\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 + 1.51i)T \)
good5 \( 1 + (-0.227 - 0.393i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.89 - 5.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.88T + 13T^{2} \)
17 \( 1 + (1.45 - 2.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.94 - 5.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.45 - 2.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 + (-2.16 + 3.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.85 - 6.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.58T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (2.45 + 4.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.56 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.896 - 1.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.33 - 4.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.94 + 6.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.909T + 71T^{2} \)
73 \( 1 + (2.60 - 4.50i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.97T + 83T^{2} \)
89 \( 1 + (-2.45 - 4.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.79T + 97T^{2} \)
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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

−9.785981223225500121685094695263, −8.490840856432082803919526320442, −8.034732707245958435720660956336, −7.28948681179023723582834351268, −6.43852517019299449924418547563, −5.47195186634409487017312590506, −4.42841071057211616944751997198, −3.54934154329015382784704680719, −2.16700877559804025459939860748, −1.30110506235240457131452202683, 1.01234541091767348470761538010, 2.60743521806491171005657578350, 3.37010504014159317806173084328, 4.60584422703375493587750472751, 5.40976587995297189364532668343, 6.03750923824808968850591244553, 7.32537941560863800329177847848, 8.282554141528754153167692418820, 8.807075453622943595704702700952, 9.291157716839017695983965186418

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