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Properties

Label 2-143-13.12-c3-0-8
Degree $2$
Conductor $143$
Sign $0.761 + 0.648i$
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  − 3.98i·2-s − 8.48·3-s − 7.85·4-s − 2.25i·5-s + 33.7i·6-s + 28.5i·7-s − 0.583i·8-s + 45.0·9-s − 8.99·10-s − 11i·11-s + 66.6·12-s + (−30.3 + 35.6i)13-s + 113.·14-s + 19.1i·15-s − 65.1·16-s + 120.·17-s + ⋯
L(s)  = 1  − 1.40i·2-s − 1.63·3-s − 0.981·4-s − 0.201i·5-s + 2.29i·6-s + 1.54i·7-s − 0.0257i·8-s + 1.66·9-s − 0.284·10-s − 0.301i·11-s + 1.60·12-s + (−0.648 + 0.761i)13-s + 2.17·14-s + 0.329i·15-s − 1.01·16-s + 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.761 + 0.648i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ 0.761 + 0.648i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.760332 - 0.279741i\)
\(L(\frac12)\) \(\approx\) \(0.760332 - 0.279741i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + 11iT \)
13 \( 1 + (30.3 - 35.6i)T \)
good2 \( 1 + 3.98iT - 8T^{2} \)
3 \( 1 + 8.48T + 27T^{2} \)
5 \( 1 + 2.25iT - 125T^{2} \)
7 \( 1 - 28.5iT - 343T^{2} \)
17 \( 1 - 120.T + 4.91e3T^{2} \)
19 \( 1 + 94.2iT - 6.85e3T^{2} \)
23 \( 1 + 92.8T + 1.21e4T^{2} \)
29 \( 1 - 156.T + 2.43e4T^{2} \)
31 \( 1 - 314. iT - 2.97e4T^{2} \)
37 \( 1 - 127. iT - 5.06e4T^{2} \)
41 \( 1 - 31.1iT - 6.89e4T^{2} \)
43 \( 1 - 447.T + 7.95e4T^{2} \)
47 \( 1 - 188. iT - 1.03e5T^{2} \)
53 \( 1 - 485.T + 1.48e5T^{2} \)
59 \( 1 - 201. iT - 2.05e5T^{2} \)
61 \( 1 + 15.5T + 2.26e5T^{2} \)
67 \( 1 + 170. iT - 3.00e5T^{2} \)
71 \( 1 + 7.43iT - 3.57e5T^{2} \)
73 \( 1 - 779. iT - 3.89e5T^{2} \)
79 \( 1 + 888.T + 4.93e5T^{2} \)
83 \( 1 - 583. iT - 5.71e5T^{2} \)
89 \( 1 + 297. iT - 7.04e5T^{2} \)
97 \( 1 - 836. iT - 9.12e5T^{2} \)
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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

−12.11251836739927543703647387033, −11.80236895746107592886897133073, −10.77066186651674829285737445776, −9.867067080890036972499636792636, −8.820791621169353648383174271412, −6.81706943123315001323258183662, −5.60502128877701152225708300979, −4.68129456214885778163235568193, −2.76211554161946493589882531017, −1.08474531291602469502790453967, 0.62866048271066784095279536431, 4.20807675645982664400449822345, 5.40037486251271498841454512718, 6.17661468854812332075763186250, 7.32032461110767339592047705505, 7.78863512981266011788746591068, 10.02756463546067254916443746161, 10.51668602099671395933432834803, 11.78543494184240111852257766741, 12.73424801377646042931064072898

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