Properties

Label 2-143-13.9-c3-0-23
Degree $2$
Conductor $143$
Sign $0.537 + 0.843i$
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  + (−2.06 + 3.58i)2-s + (2.73 − 4.74i)3-s + (−4.56 − 7.90i)4-s − 3.23·5-s + (11.3 + 19.6i)6-s + (−1.53 − 2.66i)7-s + 4.65·8-s + (−1.50 − 2.61i)9-s + (6.68 − 11.5i)10-s + (−5.5 + 9.52i)11-s − 49.9·12-s + (9.83 − 45.8i)13-s + 12.7·14-s + (−8.85 + 15.3i)15-s + (26.8 − 46.5i)16-s + (−54.0 − 93.6i)17-s + ⋯
L(s)  = 1  + (−0.731 + 1.26i)2-s + (0.527 − 0.913i)3-s + (−0.570 − 0.987i)4-s − 0.289·5-s + (0.771 + 1.33i)6-s + (−0.0829 − 0.143i)7-s + 0.205·8-s + (−0.0558 − 0.0967i)9-s + (0.211 − 0.366i)10-s + (−0.150 + 0.261i)11-s − 1.20·12-s + (0.209 − 0.977i)13-s + 0.242·14-s + (−0.152 + 0.264i)15-s + (0.419 − 0.727i)16-s + (−0.771 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.537 + 0.843i$
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: \((2,\ 143,\ (\ :3/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.713201 - 0.391087i\)
\(L(\frac12)\) \(\approx\) \(0.713201 - 0.391087i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.83 + 45.8i)T \)
good2 \( 1 + (2.06 - 3.58i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-2.73 + 4.74i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 3.23T + 125T^{2} \)
7 \( 1 + (1.53 + 2.66i)T + (-171.5 + 297. i)T^{2} \)
17 \( 1 + (54.0 + 93.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (46.1 + 79.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-55.2 + 95.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (3.91 - 6.77i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 96.9T + 2.97e4T^{2} \)
37 \( 1 + (-117. + 202. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (216. - 375. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (66.4 + 115. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 415.T + 1.03e5T^{2} \)
53 \( 1 - 69.7T + 1.48e5T^{2} \)
59 \( 1 + (0.664 + 1.15i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (121. + 210. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (136. - 237. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-540. - 936. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 99.5T + 3.89e5T^{2} \)
79 \( 1 + 1.17e3T + 4.93e5T^{2} \)
83 \( 1 - 468.T + 5.71e5T^{2} \)
89 \( 1 + (-515. + 893. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-490. - 849. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.84171420242380331761378093335, −11.48626431242921122964464969436, −10.06425344236417129449884732296, −8.842675790309874021263414847109, −8.072978322196676782107889912745, −7.23318784678393310009227410879, −6.51330212296247444690699022947, −4.93903636972374893058306192926, −2.68466591081270630318802494449, −0.47340138809887278005486330750, 1.78121828874429314165694940374, 3.38346302957459844029436658181, 4.22325664928645530443279428232, 6.26481057644475155820882147092, 8.202829450889353110861287643967, 8.955983111795162905598020283852, 9.818624554224722958608338681725, 10.61772321636864955390285318599, 11.52774048995772211166151403975, 12.45314795443819358289447723835

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