Properties

Degree 2
Conductor 29
Sign $0.625 - 0.780i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.62 + 2.62i)2-s + (−3.11 − 3.11i)3-s + 9.78i·4-s − 4.53i·5-s − 16.3i·6-s − 0.745·7-s + (−15.1 + 15.1i)8-s + 10.3i·9-s + (11.8 − 11.8i)10-s + (0.0423 + 0.0423i)11-s + (30.4 − 30.4i)12-s + 8.30i·13-s + (−1.95 − 1.95i)14-s + (−14.0 + 14.0i)15-s − 40.5·16-s + (15.2 + 15.2i)17-s + ⋯
L(s)  = 1  + (1.31 + 1.31i)2-s + (−1.03 − 1.03i)3-s + 2.44i·4-s − 0.906i·5-s − 2.72i·6-s − 0.106·7-s + (−1.89 + 1.89i)8-s + 1.15i·9-s + (1.18 − 1.18i)10-s + (0.00384 + 0.00384i)11-s + (2.53 − 2.53i)12-s + 0.638i·13-s + (−0.139 − 0.139i)14-s + (−0.939 + 0.939i)15-s − 2.53·16-s + (0.897 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.625 - 0.780i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (12, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 29,\ (\ :1),\ 0.625 - 0.780i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.17947 + 0.566144i\)
\(L(\frac12)\)  \(\approx\)  \(1.17947 + 0.566144i\)
\(L(2)\)   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 \( 1 + (-22.0 + 18.7i)T \)
good2 \( 1 + (-2.62 - 2.62i)T + 4iT^{2} \)
3 \( 1 + (3.11 + 3.11i)T + 9iT^{2} \)
5 \( 1 + 4.53iT - 25T^{2} \)
7 \( 1 + 0.745T + 49T^{2} \)
11 \( 1 + (-0.0423 - 0.0423i)T + 121iT^{2} \)
13 \( 1 - 8.30iT - 169T^{2} \)
17 \( 1 + (-15.2 - 15.2i)T + 289iT^{2} \)
19 \( 1 + (25.1 + 25.1i)T + 361iT^{2} \)
23 \( 1 - 8.64T + 529T^{2} \)
31 \( 1 + (5.09 + 5.09i)T + 961iT^{2} \)
37 \( 1 + (-10.0 + 10.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-2.50 + 2.50i)T - 1.68e3iT^{2} \)
43 \( 1 + (11.2 + 11.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (23.0 - 23.0i)T - 2.20e3iT^{2} \)
53 \( 1 + 42.8T + 2.80e3T^{2} \)
59 \( 1 - 106.T + 3.48e3T^{2} \)
61 \( 1 + (42.3 + 42.3i)T + 3.72e3iT^{2} \)
67 \( 1 - 75.7iT - 4.48e3T^{2} \)
71 \( 1 - 71.2iT - 5.04e3T^{2} \)
73 \( 1 + (73.7 - 73.7i)T - 5.32e3iT^{2} \)
79 \( 1 + (78.7 + 78.7i)T + 6.24e3iT^{2} \)
83 \( 1 - 15.1T + 6.88e3T^{2} \)
89 \( 1 + (22.8 + 22.8i)T + 7.92e3iT^{2} \)
97 \( 1 + (-42.8 + 42.8i)T - 9.40e3iT^{2} \)
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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

−16.99952974484881863574139918421, −16.00049277620544984180669140048, −14.59168759056489398741254473071, −13.08327757038801006960131246071, −12.72551103291777718119671978392, −11.56175081813304305707853331337, −8.444407355063947988627836793240, −6.97323286213651169553456736135, −5.96827886354922175232360218838, −4.64144976502120482743189231835, 3.32172985133567089676984978740, 4.89520948468533016460411422647, 6.16609247298252896762696064335, 10.02815282008404774869610468227, 10.59178990238765604463886484887, 11.57957881352770016883215074148, 12.68292357972146377991168409744, 14.31782902983355190523849846866, 15.10938512165633292155529386112, 16.46933018907083941855576230510

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