Properties

Degree 2
Conductor 29
Sign $0.646 + 0.762i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0310 − 0.0108i)2-s + (0.529 − 4.69i)3-s + (−3.12 − 2.49i)4-s + (3.79 + 7.87i)5-s + (−0.0675 + 0.140i)6-s + (3.62 + 4.54i)7-s + (0.140 + 0.222i)8-s + (−13.0 − 2.97i)9-s + (−0.0321 − 0.285i)10-s + (3.95 − 6.29i)11-s + (−13.3 + 13.3i)12-s + (−10.7 + 2.45i)13-s + (−0.0632 − 0.180i)14-s + (38.9 − 13.6i)15-s + (3.55 + 15.5i)16-s + (−4.26 − 4.26i)17-s + ⋯
L(s)  = 1  + (−0.0155 − 0.00543i)2-s + (0.176 − 1.56i)3-s + (−0.781 − 0.623i)4-s + (0.758 + 1.57i)5-s + (−0.0112 + 0.0233i)6-s + (0.518 + 0.649i)7-s + (0.0175 + 0.0278i)8-s + (−1.44 − 0.330i)9-s + (−0.00321 − 0.0285i)10-s + (0.359 − 0.572i)11-s + (−1.11 + 1.11i)12-s + (−0.827 + 0.188i)13-s + (−0.00451 − 0.0129i)14-s + (2.59 − 0.909i)15-s + (0.222 + 0.974i)16-s + (−0.250 − 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.646 + 0.762i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.646 + 0.762i)$
$L(\frac{3}{2})$  $\approx$  $0.882188 - 0.408461i$
$L(\frac12)$  $\approx$  $0.882188 - 0.408461i$
$L(2)$   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 \( 1 + (-28.9 - 0.0339i)T \)
good2 \( 1 + (0.0310 + 0.0108i)T + (3.12 + 2.49i)T^{2} \)
3 \( 1 + (-0.529 + 4.69i)T + (-8.77 - 2.00i)T^{2} \)
5 \( 1 + (-3.79 - 7.87i)T + (-15.5 + 19.5i)T^{2} \)
7 \( 1 + (-3.62 - 4.54i)T + (-10.9 + 47.7i)T^{2} \)
11 \( 1 + (-3.95 + 6.29i)T + (-52.4 - 109. i)T^{2} \)
13 \( 1 + (10.7 - 2.45i)T + (152. - 73.3i)T^{2} \)
17 \( 1 + (4.26 + 4.26i)T + 289iT^{2} \)
19 \( 1 + (13.0 - 1.47i)T + (351. - 80.3i)T^{2} \)
23 \( 1 + (6.32 + 3.04i)T + (329. + 413. i)T^{2} \)
31 \( 1 + (-2.37 - 0.830i)T + (751. + 599. i)T^{2} \)
37 \( 1 + (17.7 + 28.2i)T + (-593. + 1.23e3i)T^{2} \)
41 \( 1 + (2.70 - 2.70i)T - 1.68e3iT^{2} \)
43 \( 1 + (11.9 + 34.2i)T + (-1.44e3 + 1.15e3i)T^{2} \)
47 \( 1 + (-7.75 - 4.87i)T + (958. + 1.99e3i)T^{2} \)
53 \( 1 + (-39.1 + 18.8i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 - 70.7T + 3.48e3T^{2} \)
61 \( 1 + (11.5 - 102. i)T + (-3.62e3 - 828. i)T^{2} \)
67 \( 1 + (-26.2 - 5.98i)T + (4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (93.3 - 21.3i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (83.5 - 29.2i)T + (4.16e3 - 3.32e3i)T^{2} \)
79 \( 1 + (-51.8 + 32.5i)T + (2.70e3 - 5.62e3i)T^{2} \)
83 \( 1 + (-18.9 + 23.7i)T + (-1.53e3 - 6.71e3i)T^{2} \)
89 \( 1 + (-97.6 - 34.1i)T + (6.19e3 + 4.93e3i)T^{2} \)
97 \( 1 + (12.7 + 113. i)T + (-9.17e3 + 2.09e3i)T^{2} \)
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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

−17.55912938660836792079100994448, −14.82415151032648075597395111760, −14.22633233607128244139216430727, −13.36855423785653431879019764929, −11.82188454123490988623267279980, −10.33316715813585999122850354103, −8.674450801252874911549529204007, −6.97849959575409640813168632447, −5.86659638478132399528909409901, −2.24269937766881496962787346783, 4.31667147850594059128176818069, 4.96176256635649119705681657636, 8.318747955534313278882148975104, 9.312003220736615731492874376453, 10.18145537920618423087948833642, 12.22781640804379906531562570914, 13.47055654202820883989047603777, 14.64586094824554734570096560348, 16.15029921786494433462358738762, 17.17232785354532584056762586689

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