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Properties

Label 2-143-13.12-c3-0-13
Degree $2$
Conductor $143$
Sign $0.702 - 0.711i$
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  + 4.83i·2-s − 7.08·3-s − 15.3·4-s − 9.07i·5-s − 34.2i·6-s + 2.15i·7-s − 35.4i·8-s + 23.2·9-s + 43.8·10-s − 11i·11-s + 108.·12-s + (33.3 + 32.9i)13-s − 10.4·14-s + 64.3i·15-s + 48.4·16-s − 49.8·17-s + ⋯
L(s)  = 1  + 1.70i·2-s − 1.36·3-s − 1.91·4-s − 0.811i·5-s − 2.32i·6-s + 0.116i·7-s − 1.56i·8-s + 0.860·9-s + 1.38·10-s − 0.301i·11-s + 2.61·12-s + (0.711 + 0.702i)13-s − 0.198·14-s + 1.10i·15-s + 0.756·16-s − 0.711·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $0.702 - 0.711i$
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: \((2,\ 143,\ (\ :3/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.674124 + 0.281967i\)
\(L(\frac12)\) \(\approx\) \(0.674124 + 0.281967i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-33.3 - 32.9i)T \)
good2 \( 1 - 4.83iT - 8T^{2} \)
3 \( 1 + 7.08T + 27T^{2} \)
5 \( 1 + 9.07iT - 125T^{2} \)
7 \( 1 - 2.15iT - 343T^{2} \)
17 \( 1 + 49.8T + 4.91e3T^{2} \)
19 \( 1 + 95.3iT - 6.85e3T^{2} \)
23 \( 1 - 96.0T + 1.21e4T^{2} \)
29 \( 1 - 115.T + 2.43e4T^{2} \)
31 \( 1 + 78.6iT - 2.97e4T^{2} \)
37 \( 1 + 13.4iT - 5.06e4T^{2} \)
41 \( 1 - 122. iT - 6.89e4T^{2} \)
43 \( 1 + 95.5T + 7.95e4T^{2} \)
47 \( 1 + 361. iT - 1.03e5T^{2} \)
53 \( 1 - 617.T + 1.48e5T^{2} \)
59 \( 1 + 723. iT - 2.05e5T^{2} \)
61 \( 1 + 549.T + 2.26e5T^{2} \)
67 \( 1 + 554. iT - 3.00e5T^{2} \)
71 \( 1 + 99.2iT - 3.57e5T^{2} \)
73 \( 1 + 609. iT - 3.89e5T^{2} \)
79 \( 1 + 586.T + 4.93e5T^{2} \)
83 \( 1 - 5.23iT - 5.71e5T^{2} \)
89 \( 1 + 789. iT - 7.04e5T^{2} \)
97 \( 1 - 1.65e3iT - 9.12e5T^{2} \)
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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

−13.04695865771930109490836850701, −11.80154260718933042766805958668, −10.85574843323790638190880650972, −9.159275783786022739362419583801, −8.520975801139652047196573712292, −6.97892894247563594943551795692, −6.28439756274940341346209702116, −5.22486049313722991914378643024, −4.52264676341933255034747402166, −0.55627688215338095153922169149, 1.07854548727815336918077989682, 2.89271824540548132667926627147, 4.28644485750058674695343799626, 5.64119636893547155769752581949, 6.91722667846623213097825355049, 8.731584265632621585460344070737, 10.26439950234285349369385673143, 10.59794153758252806960850913305, 11.37639150841162724294874445763, 12.20570239873070981926679789010

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