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Properties

Label 2-143-143.32-c1-0-8
Degree $2$
Conductor $143$
Sign $0.385 + 0.922i$
Analytic cond. $1.14186$
Root an. cond. $1.06857$
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  + (−0.946 + 0.253i)2-s + (1.28 − 2.23i)3-s + (−0.901 + 0.520i)4-s + (0.744 − 0.744i)5-s + (−0.653 + 2.43i)6-s + (0.916 + 0.245i)7-s + (2.10 − 2.10i)8-s + (−1.82 − 3.15i)9-s + (−0.515 + 0.892i)10-s + (−0.657 − 3.25i)11-s + 2.68i·12-s + (−2.47 − 2.61i)13-s − 0.929·14-s + (−0.702 − 2.61i)15-s + (−0.418 + 0.724i)16-s + (2.16 + 3.74i)17-s + ⋯
L(s)  = 1  + (−0.669 + 0.179i)2-s + (0.743 − 1.28i)3-s + (−0.450 + 0.260i)4-s + (0.332 − 0.332i)5-s + (−0.266 + 0.995i)6-s + (0.346 + 0.0927i)7-s + (0.744 − 0.744i)8-s + (−0.606 − 1.05i)9-s + (−0.163 + 0.282i)10-s + (−0.198 − 0.980i)11-s + 0.774i·12-s + (−0.687 − 0.726i)13-s − 0.248·14-s + (−0.181 − 0.676i)15-s + (−0.104 + 0.181i)16-s + (0.523 + 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.385 + 0.922i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776091 - 0.516950i\)
\(L(\frac12)\) \(\approx\) \(0.776091 - 0.516950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.657 + 3.25i)T \)
13 \( 1 + (2.47 + 2.61i)T \)
good2 \( 1 + (0.946 - 0.253i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-1.28 + 2.23i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.744 + 0.744i)T - 5iT^{2} \)
7 \( 1 + (-0.916 - 0.245i)T + (6.06 + 3.5i)T^{2} \)
17 \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.160 - 0.599i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.83 - 1.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.63 - 3.25i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.29 - 1.29i)T - 31iT^{2} \)
37 \( 1 + (8.89 - 2.38i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.215 + 0.0576i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.59 - 4.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.34 - 6.34i)T + 47iT^{2} \)
53 \( 1 + 5.03T + 53T^{2} \)
59 \( 1 + (-3.10 + 11.5i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.60 + 1.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.867 + 3.23i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-9.18 - 2.46i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.237 + 0.237i)T - 73iT^{2} \)
79 \( 1 - 16.4iT - 79T^{2} \)
83 \( 1 + (8.82 + 8.82i)T + 83iT^{2} \)
89 \( 1 + (-2.32 + 0.621i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.72 + 1.53i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.91290542240460470597364209604, −12.40141567391693744305419951763, −10.73569226989187690633217904078, −9.458576369166769813098142203368, −8.386057793627166807445280198113, −8.001527185870646827685835430685, −6.86831926543422599327624881492, −5.28228092868266543263176724343, −3.21773084556825237563448577522, −1.30379968178044567012172721959, 2.43024488912653536883196508578, 4.31398474985714670295388416219, 5.10026122276581695544137086741, 7.21184613639087865255927177096, 8.515365873327626733035851233178, 9.402506371834806569499986477052, 10.02154396909203123455190308944, 10.68910699593869661170436100275, 12.10100076278759943689068575068, 13.79598882028214717315801779559

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