Properties

Degree 2
Conductor 29
Sign $0.549 - 0.835i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.277 + 1.21i)2-s + (−1.62 + 0.781i)3-s + (0.400 + 0.193i)4-s + (0.900 − 3.94i)5-s + (−0.5 − 2.19i)6-s + (−0.623 + 0.300i)7-s + (−1.90 + 2.38i)8-s + (0.153 − 0.193i)9-s + (4.54 + 2.19i)10-s + (−1.77 − 2.22i)11-s − 0.801·12-s + (0.914 + 1.14i)13-s + (−0.192 − 0.841i)14-s + (1.62 + 7.11i)15-s + (−1.81 − 2.27i)16-s − 1.60·17-s + ⋯
L(s)  = 1  + (−0.196 + 0.859i)2-s + (−0.937 + 0.451i)3-s + (0.200 + 0.0965i)4-s + (0.402 − 1.76i)5-s + (−0.204 − 0.894i)6-s + (−0.235 + 0.113i)7-s + (−0.672 + 0.842i)8-s + (0.0513 − 0.0643i)9-s + (1.43 + 0.692i)10-s + (−0.535 − 0.672i)11-s − 0.231·12-s + (0.253 + 0.318i)13-s + (−0.0513 − 0.224i)14-s + (0.419 + 1.83i)15-s + (−0.453 − 0.569i)16-s − 0.388·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.549 - 0.835i$
motivic weight  =  \(1\)
character  :  $\chi_{29} (25, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1/2),\ 0.549 - 0.835i)$
$L(1)$  $\approx$  $0.492380 + 0.265566i$
$L(\frac12)$  $\approx$  $0.492380 + 0.265566i$
$L(\frac{3}{2})$   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 \( 1 + (-3.71 - 3.89i)T \)
good2 \( 1 + (0.277 - 1.21i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (1.62 - 0.781i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 + (-0.900 + 3.94i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (0.623 - 0.300i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (1.77 + 2.22i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.914 - 1.14i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 1.60T + 17T^{2} \)
19 \( 1 + (-2.42 - 1.16i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.14 - 5.02i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (-0.434 + 1.90i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-1.77 + 2.22i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 + (0.147 + 0.648i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (2.96 + 3.71i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.0108 - 0.0476i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 6.39T + 59T^{2} \)
61 \( 1 + (-1.17 + 0.567i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-9.32 + 11.6i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (1.40 + 1.76i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (1.85 + 8.11i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (6.07 - 7.61i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-3.62 - 1.74i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (2.50 - 10.9i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-4.11 - 1.98i)T + (60.4 + 75.8i)T^{2} \)
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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

−16.98402567822991773204655584388, −16.25203566549428088037148208792, −15.78026043735343791501662921464, −13.70742248978829432316592761436, −12.34594035035493455150221024240, −11.18169602022353313051744050997, −9.316771376289671385980121419354, −8.115387743568871256004602444772, −5.99289058272781368313125466695, −5.08866170452463378308470265617, 2.78194491136502306249726834988, 6.16587374479562321953016828178, 7.02558343554993958193798692433, 9.930267503431574766805015088400, 10.78410442425223783389636846114, 11.62269724518465848846122598511, 12.93725898884036577190654568954, 14.59145313481971096415539183856, 15.74392666089394564457930219587, 17.62954068797365083169444487302

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