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Properties

Label 2-143-13.9-c3-0-10
Degree $2$
Conductor $143$
Sign $0.0334 - 0.999i$
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  + (0.856 − 1.48i)2-s + (−3.79 + 6.56i)3-s + (2.53 + 4.38i)4-s + 12.0·5-s + (6.49 + 11.2i)6-s + (−0.574 − 0.994i)7-s + 22.3·8-s + (−15.2 − 26.4i)9-s + (10.3 − 17.8i)10-s + (−5.5 + 9.52i)11-s − 38.3·12-s + (−34.0 + 32.2i)13-s − 1.96·14-s + (−45.5 + 78.9i)15-s + (−1.06 + 1.84i)16-s + (6.74 + 11.6i)17-s + ⋯
L(s)  = 1  + (0.302 − 0.524i)2-s + (−0.729 + 1.26i)3-s + (0.316 + 0.548i)4-s + 1.07·5-s + (0.442 + 0.765i)6-s + (−0.0310 − 0.0537i)7-s + 0.989·8-s + (−0.564 − 0.977i)9-s + (0.325 − 0.564i)10-s + (−0.150 + 0.261i)11-s − 0.923·12-s + (−0.725 + 0.687i)13-s − 0.0375·14-s + (−0.784 + 1.35i)15-s + (−0.0166 + 0.0288i)16-s + (0.0962 + 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0334 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0334 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.0334 - 0.999i$
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: \((2,\ 143,\ (\ :3/2),\ 0.0334 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.35926 + 1.31459i\)
\(L(\frac12)\) \(\approx\) \(1.35926 + 1.31459i\)
\(L(\frac{5}{2})\) 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 \( 1 + (34.0 - 32.2i)T \)
good2 \( 1 + (-0.856 + 1.48i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (3.79 - 6.56i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 12.0T + 125T^{2} \)
7 \( 1 + (0.574 + 0.994i)T + (-171.5 + 297. i)T^{2} \)
17 \( 1 + (-6.74 - 11.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-48.2 - 83.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-32.7 + 56.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-50.3 + 87.2i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 259.T + 2.97e4T^{2} \)
37 \( 1 + (-25.9 + 44.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-102. + 177. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-209. - 363. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 183.T + 1.03e5T^{2} \)
53 \( 1 - 183.T + 1.48e5T^{2} \)
59 \( 1 + (-369. - 640. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (310. + 537. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-397. + 688. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (208. + 361. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 231.T + 3.89e5T^{2} \)
79 \( 1 - 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + 141.T + 5.71e5T^{2} \)
89 \( 1 + (-636. + 1.10e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (105. + 182. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.65618585422182438244386099876, −11.79560946964764438594024572939, −10.77948542543754915218798729596, −10.07403373168679711885938371149, −9.235985674494628546031507868967, −7.48070583425871045967865734110, −6.00557394452374959911535942025, −4.89318917592312296315573714942, −3.79429597387805746777986675335, −2.14112385453364722213256004040, 0.953111678265519893417878681464, 2.31978490034379194487407957988, 5.28377205723561147344608558267, 5.74721128681246520297795125115, 6.87660442753729877470492613604, 7.53928365628520556520730665673, 9.359363460872221373193038152319, 10.50614489609645040155863038459, 11.45798725211607797418781920547, 12.65711294233736976175657392856

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