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Properties

Label 2-29-29.22-c1-0-0
Degree $2$
Conductor $29$
Sign $-0.203 - 0.979i$
Analytic cond. $0.231566$
Root an. cond. $0.481213$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.21 + 0.504i)2-s + (−1.23 + 2.55i)3-s + (2.83 − 1.36i)4-s + (0.0128 + 0.0564i)5-s + (1.43 − 6.28i)6-s + (1.40 + 0.677i)7-s + (−2.02 + 1.61i)8-s + (−3.16 − 3.96i)9-s + (−0.0569 − 0.118i)10-s + (3.10 + 2.47i)11-s + 8.92i·12-s + (−0.252 + 0.316i)13-s + (−3.45 − 0.788i)14-s + (−0.160 − 0.0365i)15-s + (−0.254 + 0.319i)16-s − 5.16i·17-s + ⋯
L(s)  = 1  + (−1.56 + 0.356i)2-s + (−0.711 + 1.47i)3-s + (1.41 − 0.681i)4-s + (0.00576 + 0.0252i)5-s + (0.585 − 2.56i)6-s + (0.531 + 0.256i)7-s + (−0.716 + 0.571i)8-s + (−1.05 − 1.32i)9-s + (−0.0180 − 0.0373i)10-s + (0.936 + 0.746i)11-s + 2.57i·12-s + (−0.0700 + 0.0878i)13-s + (−0.922 − 0.210i)14-s + (−0.0413 − 0.00944i)15-s + (−0.0637 + 0.0799i)16-s − 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.203 - 0.979i$
Analytic conductor: \(0.231566\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1/2),\ -0.203 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203451 + 0.250053i\)
\(L(\frac12)\) \(\approx\) \(0.203451 + 0.250053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-5.00 - 1.98i)T \)
good2 \( 1 + (2.21 - 0.504i)T + (1.80 - 0.867i)T^{2} \)
3 \( 1 + (1.23 - 2.55i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (-0.0128 - 0.0564i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + (-1.40 - 0.677i)T + (4.36 + 5.47i)T^{2} \)
11 \( 1 + (-3.10 - 2.47i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.252 - 0.316i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + 5.16iT - 17T^{2} \)
19 \( 1 + (1.49 + 3.10i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (0.0512 - 0.224i)T + (-20.7 - 9.97i)T^{2} \)
31 \( 1 + (4.21 - 0.960i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (4.19 - 3.34i)T + (8.23 - 36.0i)T^{2} \)
41 \( 1 + 1.46iT - 41T^{2} \)
43 \( 1 + (-6.32 - 1.44i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (7.48 + 5.96i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (-0.398 - 1.74i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + 1.05T + 59T^{2} \)
61 \( 1 + (-1.38 + 2.87i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (6.77 + 8.50i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (8.38 - 10.5i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-6.83 - 1.56i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + (-3.74 + 2.98i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (7.58 - 3.65i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-1.10 + 0.252i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (-7.46 - 15.5i)T + (-60.4 + 75.8i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.41338238255538119377029671800, −16.51966580013732180299985604447, −15.66650492891986675370613627358, −14.62034729798584843773413014268, −11.78800874425923702358152154291, −10.76384712983080109045606036305, −9.672530390885445524462118391230, −8.811227184375388056023285350364, −6.83506554588557243892079035888, −4.79104683189176889556865831916, 1.44050764825646200035269621137, 6.29086570761762050849236864710, 7.63763317413984098674282371790, 8.694310641539301854260727852892, 10.63539072679986432129304053824, 11.54706288040083029696680232908, 12.67735166385740621509472834113, 14.27296832841363427727550717815, 16.56169194654478534842213462143, 17.27620017814436959048978795766

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