Properties

Label 2-143-13.12-c3-0-29
Degree $2$
Conductor $143$
Sign $-0.672 - 0.740i$
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  − 2.18i·2-s − 9.62·3-s + 3.22·4-s − 10.6i·5-s + 21.0i·6-s − 26.7i·7-s − 24.5i·8-s + 65.6·9-s − 23.3·10-s + 11i·11-s − 31.0·12-s + (−34.6 + 31.5i)13-s − 58.3·14-s + 102. i·15-s − 27.7·16-s − 123.·17-s + ⋯
L(s)  = 1  − 0.772i·2-s − 1.85·3-s + 0.403·4-s − 0.956i·5-s + 1.43i·6-s − 1.44i·7-s − 1.08i·8-s + 2.43·9-s − 0.739·10-s + 0.301i·11-s − 0.746·12-s + (−0.740 + 0.672i)13-s − 1.11·14-s + 1.77i·15-s − 0.434·16-s − 1.76·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.672 - 0.740i$
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: \((2,\ 143,\ (\ :3/2),\ -0.672 - 0.740i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.219886 + 0.496738i\)
\(L(\frac12)\) \(\approx\) \(0.219886 + 0.496738i\)
\(L(\frac{5}{2})\) 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 \( 1 + (34.6 - 31.5i)T \)
good2 \( 1 + 2.18iT - 8T^{2} \)
3 \( 1 + 9.62T + 27T^{2} \)
5 \( 1 + 10.6iT - 125T^{2} \)
7 \( 1 + 26.7iT - 343T^{2} \)
17 \( 1 + 123.T + 4.91e3T^{2} \)
19 \( 1 + 6.83iT - 6.85e3T^{2} \)
23 \( 1 - 164.T + 1.21e4T^{2} \)
29 \( 1 + 184.T + 2.43e4T^{2} \)
31 \( 1 - 90.4iT - 2.97e4T^{2} \)
37 \( 1 - 98.3iT - 5.06e4T^{2} \)
41 \( 1 - 313. iT - 6.89e4T^{2} \)
43 \( 1 - 60.0T + 7.95e4T^{2} \)
47 \( 1 + 601. iT - 1.03e5T^{2} \)
53 \( 1 + 106.T + 1.48e5T^{2} \)
59 \( 1 + 128. iT - 2.05e5T^{2} \)
61 \( 1 + 102.T + 2.26e5T^{2} \)
67 \( 1 - 186. iT - 3.00e5T^{2} \)
71 \( 1 + 736. iT - 3.57e5T^{2} \)
73 \( 1 - 814. iT - 3.89e5T^{2} \)
79 \( 1 - 171.T + 4.93e5T^{2} \)
83 \( 1 - 15.5iT - 5.71e5T^{2} \)
89 \( 1 + 878. iT - 7.04e5T^{2} \)
97 \( 1 + 750. iT - 9.12e5T^{2} \)
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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

−11.75422605213346003697788283971, −11.11215574088034308903564534500, −10.42227235676328145875166426329, −9.379168068825610360101713698500, −7.13061981896630780792683885477, −6.69832892206498975418656039350, −4.95188011284490709221797849581, −4.22956353831297430533787536785, −1.49941732756318684187410309771, −0.33537639182211852991229869482, 2.44382616022503851858901362942, 5.01376947669990166914508683599, 5.83492203258011305172588262748, 6.58739000378986512032772926609, 7.43222745882529604629894474737, 9.154314986808024232726617981635, 10.85611115140006037414827296838, 11.05930768972838564525010691009, 12.06099648880668481657787427080, 12.97466330091599098432156738448

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