Properties

Degree 2
Conductor 29
Sign $0.957 + 0.289i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.45 − 1.45i)2-s + (3.81 + 3.81i)3-s + 0.234i·4-s − 3.14i·5-s − 11.0i·6-s + 0.342·7-s + (−5.47 + 5.47i)8-s + 20.0i·9-s + (−4.57 + 4.57i)10-s + (−14.0 − 14.0i)11-s + (−0.895 + 0.895i)12-s + 11.5i·13-s + (−0.498 − 0.498i)14-s + (11.9 − 11.9i)15-s + 16.8·16-s + (−6.37 − 6.37i)17-s + ⋯
L(s)  = 1  + (−0.727 − 0.727i)2-s + (1.27 + 1.27i)3-s + 0.0587i·4-s − 0.629i·5-s − 1.84i·6-s + 0.0489·7-s + (−0.684 + 0.684i)8-s + 2.22i·9-s + (−0.457 + 0.457i)10-s + (−1.27 − 1.27i)11-s + (−0.0746 + 0.0746i)12-s + 0.885i·13-s + (−0.0355 − 0.0355i)14-s + (0.799 − 0.799i)15-s + 1.05·16-s + (−0.374 − 0.374i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.957 + 0.289i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (12, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 29,\ (\ :1),\ 0.957 + 0.289i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.909737 - 0.134355i\)
\(L(\frac12)\)  \(\approx\)  \(0.909737 - 0.134355i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 29$,\(F_p(T)\) is a polynomial of degree 2. If $p = 29$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad29 \( 1 + (-25.6 - 13.4i)T \)
good2 \( 1 + (1.45 + 1.45i)T + 4iT^{2} \)
3 \( 1 + (-3.81 - 3.81i)T + 9iT^{2} \)
5 \( 1 + 3.14iT - 25T^{2} \)
7 \( 1 - 0.342T + 49T^{2} \)
11 \( 1 + (14.0 + 14.0i)T + 121iT^{2} \)
13 \( 1 - 11.5iT - 169T^{2} \)
17 \( 1 + (6.37 + 6.37i)T + 289iT^{2} \)
19 \( 1 + (6.11 + 6.11i)T + 361iT^{2} \)
23 \( 1 - 15.8T + 529T^{2} \)
31 \( 1 + (-6.18 - 6.18i)T + 961iT^{2} \)
37 \( 1 + (-24.2 + 24.2i)T - 1.36e3iT^{2} \)
41 \( 1 + (34.9 - 34.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (15.9 + 15.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-7.57 + 7.57i)T - 2.20e3iT^{2} \)
53 \( 1 - 62.1T + 2.80e3T^{2} \)
59 \( 1 + 21.4T + 3.48e3T^{2} \)
61 \( 1 + (-13.1 - 13.1i)T + 3.72e3iT^{2} \)
67 \( 1 + 17.8iT - 4.48e3T^{2} \)
71 \( 1 - 53.0iT - 5.04e3T^{2} \)
73 \( 1 + (-5.93 + 5.93i)T - 5.32e3iT^{2} \)
79 \( 1 + (81.6 + 81.6i)T + 6.24e3iT^{2} \)
83 \( 1 + 77.1T + 6.88e3T^{2} \)
89 \( 1 + (-27.4 - 27.4i)T + 7.92e3iT^{2} \)
97 \( 1 + (-48.7 + 48.7i)T - 9.40e3iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.62099517228806175705609179239, −15.72136000467878395596886420041, −14.50117541727734921785368855736, −13.35855267806136619157828561441, −11.20054939984820005351716631560, −10.22896422082032176499841244210, −8.988729861399509031544238407192, −8.434901873588437524862968822168, −4.95593808491616919262208247534, −2.83792964823455948155975766554, 2.79299997031339127130910812733, 6.71000806915674810977560924237, 7.64691071018187867877271855946, 8.473179752651137522879711081639, 10.05980861299268372901167217529, 12.48154030736968834476833463182, 13.22989118843347676893137748484, 14.84675749336087465310528439611, 15.42230757821579236299382081544, 17.43946692470197145497167837737

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