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Properties

Label 2-143-13.9-c3-0-8
Degree $2$
Conductor $143$
Sign $-0.959 - 0.280i$
Analytic cond. $8.43727$
Root an. cond. $2.90469$
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  + (−0.944 + 1.63i)2-s + (−0.783 + 1.35i)3-s + (2.21 + 3.83i)4-s + 2.20·5-s + (−1.47 − 2.56i)6-s + (0.902 + 1.56i)7-s − 23.4·8-s + (12.2 + 21.2i)9-s + (−2.08 + 3.60i)10-s + (−5.5 + 9.52i)11-s − 6.94·12-s + (22.7 + 40.9i)13-s − 3.40·14-s + (−1.72 + 2.99i)15-s + (4.45 − 7.71i)16-s + (−49.5 − 85.8i)17-s + ⋯
L(s)  = 1  + (−0.333 + 0.578i)2-s + (−0.150 + 0.261i)3-s + (0.276 + 0.479i)4-s + 0.197·5-s + (−0.100 − 0.174i)6-s + (0.0487 + 0.0843i)7-s − 1.03·8-s + (0.454 + 0.787i)9-s + (−0.0658 + 0.114i)10-s + (−0.150 + 0.261i)11-s − 0.166·12-s + (0.486 + 0.873i)13-s − 0.0650·14-s + (−0.0297 + 0.0515i)15-s + (0.0696 − 0.120i)16-s + (−0.707 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.959 - 0.280i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ -0.959 - 0.280i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.162117 + 1.13364i\)
\(L(\frac12)\) \(\approx\) \(0.162117 + 1.13364i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (5.5 - 9.52i)T \)
13 \( 1 + (-22.7 - 40.9i)T \)
good2 \( 1 + (0.944 - 1.63i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (0.783 - 1.35i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 2.20T + 125T^{2} \)
7 \( 1 + (-0.902 - 1.56i)T + (-171.5 + 297. i)T^{2} \)
17 \( 1 + (49.5 + 85.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (8.59 + 14.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (78.0 - 135. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (25.3 - 43.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 35.0T + 2.97e4T^{2} \)
37 \( 1 + (93.9 - 162. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-140. + 242. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-152. - 263. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 314.T + 1.03e5T^{2} \)
53 \( 1 - 8.37T + 1.48e5T^{2} \)
59 \( 1 + (-111. - 193. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-344. - 596. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-89.4 + 154. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-513. - 890. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 725.T + 3.89e5T^{2} \)
79 \( 1 - 377.T + 4.93e5T^{2} \)
83 \( 1 + 705.T + 5.71e5T^{2} \)
89 \( 1 + (-521. + 903. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-700. - 1.21e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.23895691639903596861422379360, −11.90868826362819352430398469863, −11.19736371151117974741308315426, −9.825400762458810174417020427453, −8.894962480155024572452636028559, −7.68348176251330569572969731001, −6.86963246753946992348809086720, −5.52968629918946175019364992303, −4.06701536241619142765871344653, −2.24318008313697639378765624917, 0.61988189035350284793164679213, 2.11204179948638766829385705452, 3.84112727530441934047735720871, 5.81201864220388064744571455614, 6.49434836631951447904183877475, 8.099410738375898969306138203930, 9.268707636135670996499655422415, 10.33105665166690870422911118715, 10.95629705094971055374600943862, 12.17988297852095146509780464080

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