Properties

Degree 2
Conductor 29
Sign $-0.229 - 0.973i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 + 1.07i)2-s + (−2.14 + 2.14i)3-s + 1.67i·4-s + 0.488i·5-s − 4.62i·6-s + 8.09·7-s + (−6.11 − 6.11i)8-s − 0.180i·9-s + (−0.527 − 0.527i)10-s + (11.3 − 11.3i)11-s + (−3.57 − 3.57i)12-s + 10.0i·13-s + (−8.73 + 8.73i)14-s + (−1.04 − 1.04i)15-s + 6.53·16-s + (−9.69 + 9.69i)17-s + ⋯
L(s)  = 1  + (−0.539 + 0.539i)2-s + (−0.714 + 0.714i)3-s + 0.417i·4-s + 0.0977i·5-s − 0.770i·6-s + 1.15·7-s + (−0.764 − 0.764i)8-s − 0.0201i·9-s + (−0.0527 − 0.0527i)10-s + (1.02 − 1.02i)11-s + (−0.298 − 0.298i)12-s + 0.772i·13-s + (−0.623 + 0.623i)14-s + (−0.0698 − 0.0698i)15-s + 0.408·16-s + (−0.570 + 0.570i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.229 - 0.973i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 29,\ (\ :1),\ -0.229 - 0.973i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.416099 + 0.525572i\)
\(L(\frac12)\)  \(\approx\)  \(0.416099 + 0.525572i\)
\(L(2)\)   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 \( 1 + (11.8 + 26.4i)T \)
good2 \( 1 + (1.07 - 1.07i)T - 4iT^{2} \)
3 \( 1 + (2.14 - 2.14i)T - 9iT^{2} \)
5 \( 1 - 0.488iT - 25T^{2} \)
7 \( 1 - 8.09T + 49T^{2} \)
11 \( 1 + (-11.3 + 11.3i)T - 121iT^{2} \)
13 \( 1 - 10.0iT - 169T^{2} \)
17 \( 1 + (9.69 - 9.69i)T - 289iT^{2} \)
19 \( 1 + (-8.58 + 8.58i)T - 361iT^{2} \)
23 \( 1 + 14.4T + 529T^{2} \)
31 \( 1 + (-31.2 + 31.2i)T - 961iT^{2} \)
37 \( 1 + (31.2 + 31.2i)T + 1.36e3iT^{2} \)
41 \( 1 + (-19.6 - 19.6i)T + 1.68e3iT^{2} \)
43 \( 1 + (50.2 - 50.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-42.2 - 42.2i)T + 2.20e3iT^{2} \)
53 \( 1 - 16.5T + 2.80e3T^{2} \)
59 \( 1 + 14.5T + 3.48e3T^{2} \)
61 \( 1 + (45.3 - 45.3i)T - 3.72e3iT^{2} \)
67 \( 1 + 133. iT - 4.48e3T^{2} \)
71 \( 1 + 33.9iT - 5.04e3T^{2} \)
73 \( 1 + (22.1 + 22.1i)T + 5.32e3iT^{2} \)
79 \( 1 + (-9.72 + 9.72i)T - 6.24e3iT^{2} \)
83 \( 1 - 64.2T + 6.88e3T^{2} \)
89 \( 1 + (119. - 119. i)T - 7.92e3iT^{2} \)
97 \( 1 + (33.5 + 33.5i)T + 9.40e3iT^{2} \)
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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

−17.03369488206893289684073171652, −16.42615848544985181963862856027, −15.20538503209184148825337943320, −13.80216856794404944717653837437, −11.80707940159206656269259970602, −11.03333806271129422494253321037, −9.252833299027033485106856672967, −8.016473004562881158360423819147, −6.25705537520000261859133951345, −4.31615943094930326648377703674, 1.42761129586385661902147002133, 5.21431381505785316141195096004, 6.92310746468817936861513692497, 8.752114514923685142549077633149, 10.28920930877565498846540346508, 11.58492680208810089219530430202, 12.27392979537076602441020967142, 14.17833693784449049736288087217, 15.21654919546831711596534239200, 17.24146000246691357945369199896

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