Properties

Degree 2
Conductor 29
Sign $0.934 + 0.355i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.170 − 1.51i)2-s + (2.08 + 3.32i)3-s + (1.63 − 0.373i)4-s + (−5.91 − 4.71i)5-s + (4.67 − 3.72i)6-s + (−1.42 + 6.25i)7-s + (−2.85 − 8.16i)8-s + (−2.76 + 5.74i)9-s + (−6.13 + 9.75i)10-s + (−5.33 + 15.2i)11-s + (4.65 + 4.65i)12-s + (−5.47 − 11.3i)13-s + (9.72 + 1.09i)14-s + (3.32 − 29.4i)15-s + (−5.83 + 2.80i)16-s + (11.5 − 11.5i)17-s + ⋯
L(s)  = 1  + (−0.0853 − 0.757i)2-s + (0.695 + 1.10i)3-s + (0.409 − 0.0933i)4-s + (−1.18 − 0.943i)5-s + (0.778 − 0.620i)6-s + (−0.204 + 0.894i)7-s + (−0.357 − 1.02i)8-s + (−0.307 + 0.638i)9-s + (−0.613 + 0.975i)10-s + (−0.484 + 1.38i)11-s + (0.387 + 0.387i)12-s + (−0.420 − 0.874i)13-s + (0.694 + 0.0782i)14-s + (0.221 − 1.96i)15-s + (−0.364 + 0.175i)16-s + (0.679 − 0.679i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.934 + 0.355i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (14, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.934 + 0.355i)$
$L(\frac{3}{2})$  $\approx$  $1.03789 - 0.190430i$
$L(\frac12)$  $\approx$  $1.03789 - 0.190430i$
$L(2)$   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 \( 1 + (-23.5 - 16.9i)T \)
good2 \( 1 + (0.170 + 1.51i)T + (-3.89 + 0.890i)T^{2} \)
3 \( 1 + (-2.08 - 3.32i)T + (-3.90 + 8.10i)T^{2} \)
5 \( 1 + (5.91 + 4.71i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (1.42 - 6.25i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (5.33 - 15.2i)T + (-94.6 - 75.4i)T^{2} \)
13 \( 1 + (5.47 + 11.3i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (-11.5 + 11.5i)T - 289iT^{2} \)
19 \( 1 + (-12.0 - 7.60i)T + (156. + 325. i)T^{2} \)
23 \( 1 + (2.84 + 3.56i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (1.32 + 11.7i)T + (-936. + 213. i)T^{2} \)
37 \( 1 + (-2.68 - 7.68i)T + (-1.07e3 + 853. i)T^{2} \)
41 \( 1 + (-10.8 - 10.8i)T + 1.68e3iT^{2} \)
43 \( 1 + (41.8 + 4.71i)T + (1.80e3 + 411. i)T^{2} \)
47 \( 1 + (28.2 + 9.87i)T + (1.72e3 + 1.37e3i)T^{2} \)
53 \( 1 + (25.1 - 31.5i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 + 72.0T + 3.48e3T^{2} \)
61 \( 1 + (36.0 + 57.3i)T + (-1.61e3 + 3.35e3i)T^{2} \)
67 \( 1 + (40.9 - 84.9i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (-13.8 - 28.7i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (-14.9 + 132. i)T + (-5.19e3 - 1.18e3i)T^{2} \)
79 \( 1 + (-82.9 + 29.0i)T + (4.87e3 - 3.89e3i)T^{2} \)
83 \( 1 + (14.4 + 63.2i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (-10.8 - 95.9i)T + (-7.72e3 + 1.76e3i)T^{2} \)
97 \( 1 + (-63.6 + 101. i)T + (-4.08e3 - 8.47e3i)T^{2} \)
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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

−16.27786928032978428059585665541, −15.56445804063489090073599233926, −14.97248268790640603122845017417, −12.52595211796944135074245880995, −11.98308897869894555486607535857, −10.22990189289957656595392905016, −9.307830410440217446378043787475, −7.78330192206618819920287563883, −4.82816081318159380374635690053, −3.11852373935073293238531571467, 3.15619892179706629784387352715, 6.57714318545561013785244835990, 7.49544590227946357391372083788, 8.210368858493496915206305060453, 10.84497660042792958368651975350, 11.96090142151743599531325558833, 13.70035826624017966348169659412, 14.48754019761411238830945506410, 15.77587372612622335496786763516, 16.75530846681358319287215809078

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