Properties

Degree 2
Conductor 29
Sign $0.357 - 0.934i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.12 + 1.40i)2-s + (−0.0990 + 0.433i)3-s + (−0.277 − 1.21i)4-s + (0.222 − 0.279i)5-s + (−0.500 − 0.626i)6-s + (0.900 − 3.94i)7-s + (−1.22 − 0.588i)8-s + (2.52 + 1.21i)9-s + (0.143 + 0.626i)10-s + (−2.62 + 1.26i)11-s + 0.554·12-s + (−4.67 + 2.25i)13-s + (4.54 + 5.70i)14-s + (0.0990 + 0.124i)15-s + (4.44 − 2.14i)16-s + 1.10·17-s + ⋯
L(s)  = 1  + (−0.794 + 0.996i)2-s + (−0.0571 + 0.250i)3-s + (−0.138 − 0.607i)4-s + (0.0995 − 0.124i)5-s + (−0.204 − 0.255i)6-s + (0.340 − 1.49i)7-s + (−0.432 − 0.208i)8-s + (0.841 + 0.405i)9-s + (0.0452 + 0.198i)10-s + (−0.791 + 0.380i)11-s + 0.160·12-s + (−1.29 + 0.624i)13-s + (1.21 + 1.52i)14-s + (0.0255 + 0.0320i)15-s + (1.11 − 0.535i)16-s + 0.269·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.357 - 0.934i$
motivic weight  =  \(1\)
character  :  $\chi_{29} (23, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1/2),\ 0.357 - 0.934i)$
$L(1)$  $\approx$  $0.411375 + 0.283092i$
$L(\frac12)$  $\approx$  $0.411375 + 0.283092i$
$L(\frac{3}{2})$   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 \( 1 + (-4.38 + 3.12i)T \)
good2 \( 1 + (1.12 - 1.40i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.0990 - 0.433i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (-0.222 + 0.279i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (-0.900 + 3.94i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (2.62 - 1.26i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (4.67 - 2.25i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 - 1.10T + 17T^{2} \)
19 \( 1 + (0.455 + 1.99i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (2.57 + 3.23i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (3.96 - 4.97i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (-2.62 - 1.26i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 - 0.396T + 41T^{2} \)
43 \( 1 + (-3.57 - 4.48i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-7.02 + 3.38i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (2.71 - 3.40i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 9.10T + 59T^{2} \)
61 \( 1 + (-1.34 + 5.89i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (0.337 + 0.162i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-10.2 + 4.94i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (5.57 + 6.99i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-0.535 - 0.257i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (-2.09 - 9.19i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-0.887 + 1.11i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (3.50 + 15.3i)T + (-87.3 + 42.0i)T^{2} \)
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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

−17.12766126523671246419827101119, −16.51348763886378539153465335735, −15.36853698529461451713548474764, −14.07884477605949240212508926179, −12.60491504219486423836120456389, −10.57112998500381496376355919585, −9.596757290493155621182311957349, −7.75962509752452597228196744040, −7.03706179140409979728308016066, −4.67898505343392185417733475953, 2.46259427346092974642579421634, 5.66325346560503472403248174041, 7.965079999948813456966409270612, 9.389528532047503245348854437306, 10.45055346583592329462401095693, 11.97719977952906570989708688884, 12.59967987249435458310029097238, 14.68656156434386290435533914760, 15.71774688132796024959834704821, 17.62016045812302062496678613487

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