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Properties

Label 2-143-143.64-c1-0-4
Degree $2$
Conductor $143$
Sign $-0.0218 - 0.999i$
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  + (2.34 + 0.762i)2-s + (−2.33 + 1.69i)3-s + (3.30 + 2.39i)4-s + (−1.66 + 0.541i)5-s + (−6.76 + 2.19i)6-s + (1.38 − 1.91i)7-s + (3.01 + 4.15i)8-s + (1.64 − 5.04i)9-s − 4.31·10-s + (3.30 + 0.293i)11-s − 11.7·12-s + (1.33 + 3.35i)13-s + (4.71 − 3.42i)14-s + (2.96 − 4.08i)15-s + (1.39 + 4.27i)16-s + (−1.93 − 5.94i)17-s + ⋯
L(s)  = 1  + (1.65 + 0.538i)2-s + (−1.34 + 0.978i)3-s + (1.65 + 1.19i)4-s + (−0.744 + 0.242i)5-s + (−2.75 + 0.896i)6-s + (0.524 − 0.722i)7-s + (1.06 + 1.46i)8-s + (0.546 − 1.68i)9-s − 1.36·10-s + (0.996 + 0.0883i)11-s − 3.39·12-s + (0.369 + 0.929i)13-s + (1.25 − 0.915i)14-s + (0.766 − 1.05i)15-s + (0.347 + 1.06i)16-s + (−0.468 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0218 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.0218 - 0.999i$
Analytic conductor: \(1.14186\)
Root analytic conductor: \(1.06857\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :1/2),\ -0.0218 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20238 + 1.22895i\)
\(L(\frac12)\) \(\approx\) \(1.20238 + 1.22895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-3.30 - 0.293i)T \)
13 \( 1 + (-1.33 - 3.35i)T \)
good2 \( 1 + (-2.34 - 0.762i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.33 - 1.69i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.66 - 0.541i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.38 + 1.91i)T + (-2.16 - 6.65i)T^{2} \)
17 \( 1 + (1.93 + 5.94i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.777 - 1.06i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.54T + 23T^{2} \)
29 \( 1 + (7.61 + 5.53i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.363 - 0.118i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.37 - 1.89i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.10 + 7.02i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.19T + 43T^{2} \)
47 \( 1 + (-4.67 - 6.42i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.27 - 7.01i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.28 + 10.0i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.970 - 2.98i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 7.31iT - 67T^{2} \)
71 \( 1 + (-6.63 + 2.15i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.947 - 1.30i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.250 + 0.771i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (7.14 - 2.32i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 4.66iT - 89T^{2} \)
97 \( 1 + (4.95 + 1.60i)T + (78.4 + 57.0i)T^{2} \)
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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

−13.59220266266219095644150257238, −12.08446367178666442125701468835, −11.49312258814898677406581881664, −11.08288806330567401330021100135, −9.464931197726115842075809983424, −7.36912981718390909803592797171, −6.58011240942722698238127118621, −5.38220572958839038165684810891, −4.32923870725042829314284532197, −3.84242431148999192960060946927, 1.67721587851046545190457218661, 3.79597796815293413174016561846, 5.15139645087005101737000342950, 5.91447646563176277095940875802, 6.89581227887906077451301125698, 8.379087913850610684843263567537, 10.70090212368565247354596866815, 11.42848866835440610300260545027, 11.96686260065015145230137230927, 12.71527560880094647466104200844

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