Properties

Degree 2
Conductor 29
Sign $-0.435 - 0.900i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.388 + 3.45i)2-s + (−0.327 + 0.521i)3-s + (−7.86 − 1.79i)4-s + (4.44 − 3.54i)5-s + (−1.67 − 1.33i)6-s + (1.25 + 5.49i)7-s + (4.66 − 13.3i)8-s + (3.74 + 7.76i)9-s + (10.5 + 16.7i)10-s + (−6.29 − 17.9i)11-s + (3.51 − 3.51i)12-s + (6.68 − 13.8i)13-s + (−19.4 + 2.19i)14-s + (0.392 + 3.48i)15-s + (15.1 + 7.28i)16-s + (1.08 + 1.08i)17-s + ⋯
L(s)  = 1  + (−0.194 + 1.72i)2-s + (−0.109 + 0.173i)3-s + (−1.96 − 0.448i)4-s + (0.889 − 0.709i)5-s + (−0.279 − 0.222i)6-s + (0.179 + 0.784i)7-s + (0.583 − 1.66i)8-s + (0.415 + 0.862i)9-s + (1.05 + 1.67i)10-s + (−0.572 − 1.63i)11-s + (0.292 − 0.292i)12-s + (0.514 − 1.06i)13-s + (−1.38 + 0.156i)14-s + (0.0261 + 0.232i)15-s + (0.946 + 0.455i)16-s + (0.0636 + 0.0636i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.435 - 0.900i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (27, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ -0.435 - 0.900i)$
$L(\frac{3}{2})$  $\approx$  $0.471774 + 0.752501i$
$L(\frac12)$  $\approx$  $0.471774 + 0.752501i$
$L(2)$   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 \( 1 + (18.9 + 21.9i)T \)
good2 \( 1 + (0.388 - 3.45i)T + (-3.89 - 0.890i)T^{2} \)
3 \( 1 + (0.327 - 0.521i)T + (-3.90 - 8.10i)T^{2} \)
5 \( 1 + (-4.44 + 3.54i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 + (-1.25 - 5.49i)T + (-44.1 + 21.2i)T^{2} \)
11 \( 1 + (6.29 + 17.9i)T + (-94.6 + 75.4i)T^{2} \)
13 \( 1 + (-6.68 + 13.8i)T + (-105. - 132. i)T^{2} \)
17 \( 1 + (-1.08 - 1.08i)T + 289iT^{2} \)
19 \( 1 + (17.6 - 11.1i)T + (156. - 325. i)T^{2} \)
23 \( 1 + (-4.81 + 6.03i)T + (-117. - 515. i)T^{2} \)
31 \( 1 + (5.79 - 51.4i)T + (-936. - 213. i)T^{2} \)
37 \( 1 + (0.580 - 1.65i)T + (-1.07e3 - 853. i)T^{2} \)
41 \( 1 + (-0.198 + 0.198i)T - 1.68e3iT^{2} \)
43 \( 1 + (16.5 - 1.86i)T + (1.80e3 - 411. i)T^{2} \)
47 \( 1 + (2.91 - 1.02i)T + (1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (-31.3 - 39.3i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 + 1.15T + 3.48e3T^{2} \)
61 \( 1 + (12.3 - 19.7i)T + (-1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (-2.18 - 4.54i)T + (-2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (-12.0 + 24.9i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (8.52 + 75.7i)T + (-5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (-27.2 - 9.54i)T + (4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (5.13 - 22.4i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (-8.12 + 72.1i)T + (-7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (-65.8 - 104. i)T + (-4.08e3 + 8.47e3i)T^{2} \)
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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

−16.90140961889483361326446842611, −16.21822511461988588768414957927, −15.24335799678040407720118255835, −13.79649434079854710558472684927, −13.01547251518227172837943518556, −10.53482253411886460630680568965, −8.853885241914264640793125193824, −8.084459185568106659768830926449, −5.93359224129102243340300101397, −5.26488221326253593923376502363, 2.00214759199398563929312893731, 4.18086129816790830092916897276, 6.92000484818739799004329756026, 9.377455795078679072933931214794, 10.19106768090558133909854524346, 11.28040615644736813623144456571, 12.66080540491829546099210344697, 13.55365709182353757379568503529, 14.91499086483421469339526971428, 17.26496603594466093723778057240

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