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Properties

Label 2-29-29.12-c10-0-22
Degree $2$
Conductor $29$
Sign $-0.924 - 0.380i$
Analytic cond. $18.4253$
Root an. cond. $4.29247$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (21.5 + 21.5i)2-s + (−331. − 331. i)3-s − 95.4i·4-s − 4.92e3i·5-s − 1.42e4i·6-s − 7.13e3·7-s + (2.41e4 − 2.41e4i)8-s + 1.60e5i·9-s + (1.06e5 − 1.06e5i)10-s + (−1.03e5 − 1.03e5i)11-s + (−3.16e4 + 3.16e4i)12-s − 8.24e4i·13-s + (−1.53e5 − 1.53e5i)14-s + (−1.63e6 + 1.63e6i)15-s + 9.41e5·16-s + (9.28e5 + 9.28e5i)17-s + ⋯
L(s)  = 1  + (0.673 + 0.673i)2-s + (−1.36 − 1.36i)3-s − 0.0931i·4-s − 1.57i·5-s − 1.83i·6-s − 0.424·7-s + (0.736 − 0.736i)8-s + 2.71i·9-s + (1.06 − 1.06i)10-s + (−0.641 − 0.641i)11-s + (−0.127 + 0.127i)12-s − 0.222i·13-s + (−0.285 − 0.285i)14-s + (−2.14 + 2.14i)15-s + 0.898·16-s + (0.654 + 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.924 - 0.380i$
Analytic conductor: \(18.4253\)
Root analytic conductor: \(4.29247\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5),\ -0.924 - 0.380i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.185121 + 0.936858i\)
\(L(\frac12)\) \(\approx\) \(0.185121 + 0.936858i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (1.71e7 + 1.12e7i)T \)
good2 \( 1 + (-21.5 - 21.5i)T + 1.02e3iT^{2} \)
3 \( 1 + (331. + 331. i)T + 5.90e4iT^{2} \)
5 \( 1 + 4.92e3iT - 9.76e6T^{2} \)
7 \( 1 + 7.13e3T + 2.82e8T^{2} \)
11 \( 1 + (1.03e5 + 1.03e5i)T + 2.59e10iT^{2} \)
13 \( 1 + 8.24e4iT - 1.37e11T^{2} \)
17 \( 1 + (-9.28e5 - 9.28e5i)T + 2.01e12iT^{2} \)
19 \( 1 + (1.56e6 + 1.56e6i)T + 6.13e12iT^{2} \)
23 \( 1 - 7.81e6T + 4.14e13T^{2} \)
31 \( 1 + (-3.04e7 - 3.04e7i)T + 8.19e14iT^{2} \)
37 \( 1 + (8.17e7 - 8.17e7i)T - 4.80e15iT^{2} \)
41 \( 1 + (-4.93e6 + 4.93e6i)T - 1.34e16iT^{2} \)
43 \( 1 + (6.18e7 + 6.18e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (-1.95e8 + 1.95e8i)T - 5.25e16iT^{2} \)
53 \( 1 - 3.06e8T + 1.74e17T^{2} \)
59 \( 1 - 2.77e8T + 5.11e17T^{2} \)
61 \( 1 + (-1.57e8 - 1.57e8i)T + 7.13e17iT^{2} \)
67 \( 1 + 5.60e8iT - 1.82e18T^{2} \)
71 \( 1 + 1.11e9iT - 3.25e18T^{2} \)
73 \( 1 + (-1.06e9 + 1.06e9i)T - 4.29e18iT^{2} \)
79 \( 1 + (8.72e8 + 8.72e8i)T + 9.46e18iT^{2} \)
83 \( 1 + 4.66e9T + 1.55e19T^{2} \)
89 \( 1 + (3.09e9 + 3.09e9i)T + 3.11e19iT^{2} \)
97 \( 1 + (9.13e9 - 9.13e9i)T - 7.37e19iT^{2} \)
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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

−13.46206559741102248678389771925, −13.06744913570435598119062882060, −12.11413565967894655229484356179, −10.54248639348748377937159367303, −8.270399934958996397221405030203, −6.81033087590082495946744049597, −5.63078542052286588519118558847, −4.94429297621356316153001695460, −1.32132758835334520604656169174, −0.36840406376728501515223853944, 2.92485771237024840356628059859, 4.05938224462654491658304450703, 5.48014409840850931075349924419, 7.00436117511691598786013371014, 9.849281999905145737756352455951, 10.71552616485454456630954966760, 11.44294552695154894963275406014, 12.60863715994597612360375841457, 14.44432432699818112508435125327, 15.40829352442729869151017848585

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