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Properties

Label 2-29-29.4-c1-0-1
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: $\chi_{29} (4, \cdot )$
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

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

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