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Properties

Label 2-143-13.9-c1-0-8
Degree $2$
Conductor $143$
Sign $0.533 + 0.846i$
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  + (−1.26 + 2.19i)2-s + (1.49 − 2.58i)3-s + (−2.21 − 3.83i)4-s − 3.35·5-s + (3.78 + 6.56i)6-s + (−0.959 − 1.66i)7-s + 6.14·8-s + (−2.97 − 5.14i)9-s + (4.24 − 7.35i)10-s + (0.5 − 0.866i)11-s − 13.2·12-s + (0.775 − 3.52i)13-s + 4.86·14-s + (−5.01 + 8.68i)15-s + (−3.36 + 5.82i)16-s + (0.986 + 1.70i)17-s + ⋯
L(s)  = 1  + (−0.896 + 1.55i)2-s + (0.863 − 1.49i)3-s + (−1.10 − 1.91i)4-s − 1.49·5-s + (1.54 + 2.67i)6-s + (−0.362 − 0.627i)7-s + 2.17·8-s + (−0.990 − 1.71i)9-s + (1.34 − 2.32i)10-s + (0.150 − 0.261i)11-s − 3.81·12-s + (0.215 − 0.976i)13-s + 1.29·14-s + (−1.29 + 2.24i)15-s + (−0.840 + 1.45i)16-s + (0.239 + 0.414i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.533 + 0.846i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ 0.533 + 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.495531 - 0.273451i\)
\(L(\frac12)\) \(\approx\) \(0.495531 - 0.273451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.775 + 3.52i)T \)
good2 \( 1 + (1.26 - 2.19i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.49 + 2.58i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.35T + 5T^{2} \)
7 \( 1 + (0.959 + 1.66i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (-0.986 - 1.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.219 + 0.380i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.24 - 2.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.63T + 31T^{2} \)
37 \( 1 + (-3.89 + 6.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.85 + 8.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.29 - 5.71i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.104T + 47T^{2} \)
53 \( 1 - 0.381T + 53T^{2} \)
59 \( 1 + (-0.516 - 0.894i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.86 - 4.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.62 + 6.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.63 + 9.75i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.808T + 73T^{2} \)
79 \( 1 + 7.04T + 79T^{2} \)
83 \( 1 - 8.70T + 83T^{2} \)
89 \( 1 + (5.16 - 8.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.69 - 16.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.17731272107202035268644911218, −12.26629699973215500798644033871, −10.72570741647307565436998881929, −9.148150288234965157827494596539, −8.202593682103084400157489375731, −7.63743477439071663198266203276, −7.10986251231462622101970274677, −5.89454920890482954836337561822, −3.63124628677789978180952243469, −0.71619804057158241587552221020, 2.73530454088216385348661350254, 3.71871175602936335143314039545, 4.53251942954580346336231903778, 7.64423924047791812164835816386, 8.698443762619247507583452032651, 9.256511080271123311826093492972, 10.08533615673956722460568494492, 11.33535043795859313231671257842, 11.62528448046160664061206003068, 12.89285204802552685221432586354

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