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Properties

Label 2-1344-28.19-c1-0-5
Degree $2$
Conductor $1344$
Sign $-0.0633 - 0.997i$
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 + (3 + 1.73i)5-s + (−0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (−3 + 1.73i)11-s + 5.19i·13-s − 3.46i·15-s + (−6 + 3.46i)17-s + (−3.5 + 6.06i)19-s + (−2 + 1.73i)21-s + (3.5 + 6.06i)25-s + 0.999·27-s + (2.5 + 4.33i)31-s + (3 + 1.73i)33-s + (3 − 8.66i)35-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (1.34 + 0.774i)5-s + (−0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.904 + 0.522i)11-s + 1.44i·13-s − 0.894i·15-s + (−1.45 + 0.840i)17-s + (−0.802 + 1.39i)19-s + (−0.436 + 0.377i)21-s + (0.700 + 1.21i)25-s + 0.192·27-s + (0.449 + 0.777i)31-s + (0.522 + 0.301i)33-s + (0.507 − 1.46i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.171623863\)
\(L(\frac12)\) \(\approx\) \(1.171623863\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 - 1.73i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (6 - 3.46i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.5 - 7.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−10.01358578187678467973285965681, −9.138583539104255913167845357693, −8.111294598068141072307414067493, −7.06514296176328701959695351664, −6.57688822912740895268988219936, −5.98713988355744226536585042432, −4.78941484943809288376211152268, −3.80968836216855497602501613923, −2.27417730767535000593169233888, −1.76067376641368651580485753200, 0.45446641738593379963099074030, 2.30824904945924204827088617707, 2.90449079513734976829214785036, 4.72163424407434968305944362402, 5.16019223796695592025845711692, 5.92557343419238126302057919266, 6.58943109740842429512723640410, 8.104652258271756420031806969559, 8.774647574478406259152580327013, 9.401125670394863516360453610713

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