Properties

Degree 2
Conductor 29
Sign $0.187 - 0.982i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.293 + 2.60i)2-s + (0.0782 + 0.124i)3-s + (−2.78 + 0.635i)4-s + (−1.65 − 1.31i)5-s + (−0.301 + 0.240i)6-s + (0.747 − 3.27i)7-s + (0.990 + 2.83i)8-s + (3.89 − 8.08i)9-s + (2.94 − 4.68i)10-s + (−0.536 + 1.53i)11-s + (−0.296 − 0.296i)12-s + (−3.68 − 7.65i)13-s + (8.74 + 0.985i)14-s + (0.0348 − 0.309i)15-s + (−17.3 + 8.36i)16-s + (−19.7 + 19.7i)17-s + ⋯
L(s)  = 1  + (0.146 + 1.30i)2-s + (0.0260 + 0.0415i)3-s + (−0.695 + 0.158i)4-s + (−0.330 − 0.263i)5-s + (−0.0501 + 0.0400i)6-s + (0.106 − 0.468i)7-s + (0.123 + 0.353i)8-s + (0.432 − 0.898i)9-s + (0.294 − 0.468i)10-s + (−0.0487 + 0.139i)11-s + (−0.0247 − 0.0247i)12-s + (−0.283 − 0.589i)13-s + (0.624 + 0.0703i)14-s + (0.00232 − 0.0206i)15-s + (−1.08 + 0.522i)16-s + (−1.16 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.187 - 0.982i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (14, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.187 - 0.982i)$
$L(\frac{3}{2})$  $\approx$  $0.796316 + 0.658361i$
$L(\frac12)$  $\approx$  $0.796316 + 0.658361i$
$L(2)$   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 \( 1 + (-12.6 + 26.0i)T \)
good2 \( 1 + (-0.293 - 2.60i)T + (-3.89 + 0.890i)T^{2} \)
3 \( 1 + (-0.0782 - 0.124i)T + (-3.90 + 8.10i)T^{2} \)
5 \( 1 + (1.65 + 1.31i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (-0.747 + 3.27i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (0.536 - 1.53i)T + (-94.6 - 75.4i)T^{2} \)
13 \( 1 + (3.68 + 7.65i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (19.7 - 19.7i)T - 289iT^{2} \)
19 \( 1 + (9.28 + 5.83i)T + (156. + 325. i)T^{2} \)
23 \( 1 + (-20.8 - 26.1i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (-3.96 - 35.1i)T + (-936. + 213. i)T^{2} \)
37 \( 1 + (-2.28 - 6.52i)T + (-1.07e3 + 853. i)T^{2} \)
41 \( 1 + (22.9 + 22.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (-55.6 - 6.26i)T + (1.80e3 + 411. i)T^{2} \)
47 \( 1 + (-60.5 - 21.2i)T + (1.72e3 + 1.37e3i)T^{2} \)
53 \( 1 + (22.4 - 28.0i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 + 84.6T + 3.48e3T^{2} \)
61 \( 1 + (38.2 + 60.8i)T + (-1.61e3 + 3.35e3i)T^{2} \)
67 \( 1 + (14.4 - 29.9i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (2.14 + 4.45i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (-8.36 + 74.2i)T + (-5.19e3 - 1.18e3i)T^{2} \)
79 \( 1 + (-28.5 + 9.99i)T + (4.87e3 - 3.89e3i)T^{2} \)
83 \( 1 + (7.54 + 33.0i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (-7.04 - 62.4i)T + (-7.72e3 + 1.76e3i)T^{2} \)
97 \( 1 + (21.4 - 34.0i)T + (-4.08e3 - 8.47e3i)T^{2} \)
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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

−17.19407810357676793116779691888, −15.58051578176075044659095059633, −15.26395768445614624888381283597, −13.80755555931086605263671533450, −12.48094524194685655525692939752, −10.74482572533275549917861452588, −8.883245177251376980323890190246, −7.53101644713485027683921518924, −6.27837703960268647104166312972, −4.42952859368178009667574778504, 2.45063910181647824651044181471, 4.54881365820755230050201212206, 7.12371672478975374246527783104, 9.086931595495003755612528601227, 10.62147807615503491320029471600, 11.47431448876077866308600984063, 12.69115875446537996607114865177, 13.79970641515356596209397123899, 15.40919010662463844050621850757, 16.61557437201508147535249798774

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