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Properties

Label 2-143-143.98-c1-0-6
Degree $2$
Conductor $143$
Sign $0.330 - 0.943i$
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  + (0.655 + 2.44i)2-s + (1.51 − 2.62i)3-s + (−3.81 + 2.20i)4-s + (1.00 + 1.00i)5-s + (7.40 + 1.98i)6-s + (−0.0740 + 0.276i)7-s + (−4.30 − 4.30i)8-s + (−3.08 − 5.34i)9-s + (−1.80 + 3.12i)10-s + (2.52 + 2.15i)11-s + 13.3i·12-s + (−3.39 + 1.21i)13-s − 0.723·14-s + (4.16 − 1.11i)15-s + (3.30 − 5.72i)16-s + (−3.16 − 5.47i)17-s + ⋯
L(s)  = 1  + (0.463 + 1.72i)2-s + (0.874 − 1.51i)3-s + (−1.90 + 1.10i)4-s + (0.450 + 0.450i)5-s + (3.02 + 0.810i)6-s + (−0.0279 + 0.104i)7-s + (−1.52 − 1.52i)8-s + (−1.02 − 1.78i)9-s + (−0.569 + 0.986i)10-s + (0.761 + 0.648i)11-s + 3.85i·12-s + (−0.941 + 0.335i)13-s − 0.193·14-s + (1.07 − 0.288i)15-s + (0.826 − 1.43i)16-s + (−0.767 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.330 - 0.943i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31277 + 0.930764i\)
\(L(\frac12)\) \(\approx\) \(1.31277 + 0.930764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.52 - 2.15i)T \)
13 \( 1 + (3.39 - 1.21i)T \)
good2 \( 1 + (-0.655 - 2.44i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-1.51 + 2.62i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.00 - 1.00i)T + 5iT^{2} \)
7 \( 1 + (0.0740 - 0.276i)T + (-6.06 - 3.5i)T^{2} \)
17 \( 1 + (3.16 + 5.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.21 + 0.324i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.18 + 2.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.95 - 3.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.708 + 0.708i)T + 31iT^{2} \)
37 \( 1 + (-1.89 - 7.07i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.54 + 5.74i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.41 - 9.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.64 - 4.64i)T - 47iT^{2} \)
53 \( 1 - 2.50T + 53T^{2} \)
59 \( 1 + (1.69 + 0.452i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.76 + 1.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.27 + 1.41i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.471 - 1.76i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.50 + 2.50i)T + 73iT^{2} \)
79 \( 1 + 1.84iT - 79T^{2} \)
83 \( 1 + (1.97 - 1.97i)T - 83iT^{2} \)
89 \( 1 + (-1.09 - 4.10i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.28 + 4.80i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.81870864747328089991944474137, −12.74528325919448193971404664472, −11.97740934179577158356525194104, −9.584369491876645232818911252013, −8.666539058979756278247127061674, −7.62038992320917619781319802562, −6.81547818996678790132919475279, −6.33979411268106477373668197322, −4.58378025528063773822920758791, −2.53442799029288268995493782503, 2.22312351510238397251321780676, 3.62336523889424030658719168279, 4.35536051175437867250358330235, 5.55987713566707436961065976691, 8.471756674780785309331083936294, 9.228516227170975473119494834026, 10.04120414619453904349600734659, 10.67271370189736227252616706883, 11.75450739009765767388141471611, 12.93243273560776717623352621841

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