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Properties

Label 2-143-143.25-c1-0-10
Degree $2$
Conductor $143$
Sign $-0.869 + 0.493i$
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  + (−0.571 − 0.786i)2-s + (−0.850 − 2.61i)3-s + (0.326 − 1.00i)4-s + (0.559 − 0.769i)5-s + (−1.57 + 2.16i)6-s + (3.30 + 1.07i)7-s + (−2.82 + 0.917i)8-s + (−3.70 + 2.68i)9-s − 0.924·10-s + (−2.05 + 2.60i)11-s − 2.90·12-s + (0.167 − 3.60i)13-s + (−1.04 − 3.20i)14-s + (−2.49 − 0.809i)15-s + (0.627 + 0.456i)16-s + (5.25 + 3.81i)17-s + ⋯
L(s)  = 1  + (−0.403 − 0.556i)2-s + (−0.491 − 1.51i)3-s + (0.163 − 0.501i)4-s + (0.250 − 0.344i)5-s + (−0.642 + 0.883i)6-s + (1.24 + 0.405i)7-s + (−0.998 + 0.324i)8-s + (−1.23 + 0.896i)9-s − 0.292·10-s + (−0.620 + 0.783i)11-s − 0.838·12-s + (0.0465 − 0.998i)13-s + (−0.278 − 0.857i)14-s + (−0.643 − 0.208i)15-s + (0.156 + 0.114i)16-s + (1.27 + 0.925i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.869 + 0.493i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.869 + 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.223375 - 0.845524i\)
\(L(\frac12)\) \(\approx\) \(0.223375 - 0.845524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (2.05 - 2.60i)T \)
13 \( 1 + (-0.167 + 3.60i)T \)
good2 \( 1 + (0.571 + 0.786i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.850 + 2.61i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.559 + 0.769i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-3.30 - 1.07i)T + (5.66 + 4.11i)T^{2} \)
17 \( 1 + (-5.25 - 3.81i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.17 + 0.382i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.10T + 23T^{2} \)
29 \( 1 + (-1.74 + 5.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.44 + 6.11i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-6.18 - 2.01i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.00 - 0.976i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.54T + 43T^{2} \)
47 \( 1 + (-9.74 + 3.16i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.02 - 2.20i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (9.96 + 3.23i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.95 - 5.05i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.79iT - 67T^{2} \)
71 \( 1 + (4.41 - 6.07i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.34 - 1.73i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.32 - 2.41i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.57 - 3.53i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.0413iT - 89T^{2} \)
97 \( 1 + (-6.94 - 9.55i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.46201749841627608394881216001, −11.78599934606712852007754534086, −10.86047255051308383289080184117, −9.789649747769831074440861028206, −8.235122965493897107523690483572, −7.58421614906823650619884314270, −5.93788208398092741432918160747, −5.31287312055255410809538532146, −2.27089147241062074928239131179, −1.21623288458060846358033663641, 3.27468830233629701172236928613, 4.65769321494349148956015204631, 5.76214216709244439360789108234, 7.26637510399888942702659135180, 8.383638589382948950217294292841, 9.404293479336186526881091178982, 10.51833577771706187685684255741, 11.26379976773969849902427766964, 12.14005987985653360730574562479, 14.06060126616432373523661778623

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