Properties

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

Origins

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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  :  $\chi_{29} (2, \cdot )$
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 \( 1 + (36.1 - 22.6i)T + (4.08e3 - 8.47e3i)T^{2} \)
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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

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

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