Properties

Degree 2
Conductor 29
Sign $0.974 + 0.225i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.415 − 0.0467i)2-s + (2.68 − 1.68i)3-s + (−3.72 + 0.851i)4-s + (0.738 + 0.589i)5-s + (1.03 − 0.825i)6-s + (−0.577 + 2.53i)7-s + (−3.08 + 1.07i)8-s + (0.449 − 0.932i)9-s + (0.334 + 0.209i)10-s + (−8.65 − 3.02i)11-s + (−8.56 + 8.56i)12-s + (−4.51 − 9.37i)13-s + (−0.121 + 1.07i)14-s + (2.97 + 0.335i)15-s + (12.5 − 6.04i)16-s + (21.4 + 21.4i)17-s + ⋯
L(s)  = 1  + (0.207 − 0.0233i)2-s + (0.894 − 0.561i)3-s + (−0.932 + 0.212i)4-s + (0.147 + 0.117i)5-s + (0.172 − 0.137i)6-s + (−0.0824 + 0.361i)7-s + (−0.385 + 0.134i)8-s + (0.0499 − 0.103i)9-s + (0.0334 + 0.0209i)10-s + (−0.786 − 0.275i)11-s + (−0.714 + 0.714i)12-s + (−0.347 − 0.720i)13-s + (−0.00866 + 0.0769i)14-s + (0.198 + 0.0223i)15-s + (0.784 − 0.377i)16-s + (1.26 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.974 + 0.225i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (15, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.974 + 0.225i)$
$L(\frac{3}{2})$  $\approx$  $1.11334 - 0.127015i$
$L(\frac12)$  $\approx$  $1.11334 - 0.127015i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 29$, \(F_p\) is a polynomial of degree 2. If $p = 29$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad29 \( 1 + (24.3 - 15.7i)T \)
good2 \( 1 + (-0.415 + 0.0467i)T + (3.89 - 0.890i)T^{2} \)
3 \( 1 + (-2.68 + 1.68i)T + (3.90 - 8.10i)T^{2} \)
5 \( 1 + (-0.738 - 0.589i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (0.577 - 2.53i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (8.65 + 3.02i)T + (94.6 + 75.4i)T^{2} \)
13 \( 1 + (4.51 + 9.37i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (-21.4 - 21.4i)T + 289iT^{2} \)
19 \( 1 + (-14.2 + 22.6i)T + (-156. - 325. i)T^{2} \)
23 \( 1 + (2.27 + 2.85i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (-21.7 + 2.44i)T + (936. - 213. i)T^{2} \)
37 \( 1 + (46.0 - 16.1i)T + (1.07e3 - 853. i)T^{2} \)
41 \( 1 + (25.3 - 25.3i)T - 1.68e3iT^{2} \)
43 \( 1 + (-2.78 + 24.6i)T + (-1.80e3 - 411. i)T^{2} \)
47 \( 1 + (-7.49 + 21.4i)T + (-1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (18.6 - 23.3i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 - 18.2T + 3.48e3T^{2} \)
61 \( 1 + (78.8 - 49.5i)T + (1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (-24.5 + 50.9i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (-56.1 - 116. i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (-47.2 - 5.32i)T + (5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (13.7 + 39.2i)T + (-4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (29.4 + 129. i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (-32.3 + 3.64i)T + (7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (36.1 + 22.6i)T + (4.08e3 + 8.47e3i)T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.08101444425267158682914534700, −15.31676900216299649223155591864, −14.17186740219259830427231043112, −13.30491528969641525440540663359, −12.35758688987759699301752298671, −10.22295439552180853806801587786, −8.697173760857372190145326127897, −7.76026180131083907428655894018, −5.39705689307691073367395271469, −3.05971723940551040384636244451, 3.58633049243738481376447072993, 5.25931041055721269380504203799, 7.78739878474684240259086132770, 9.341517216760252394283863023683, 9.984204537705855650000623892272, 12.17285986553491679311606352355, 13.73195394915243870995370153468, 14.27845399368517752772454627499, 15.50257035069779237635652054229, 16.87732382497038192593113041750

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