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Properties

Label 2-29-29.17-c6-0-4
Degree $2$
Conductor $29$
Sign $0.379 - 0.925i$
Analytic cond. $6.67156$
Root an. cond. $2.58293$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.72 − 4.72i)2-s + (−5.84 + 5.84i)3-s + 19.3i·4-s + 122. i·5-s + 55.2i·6-s − 317.·7-s + (393. + 393. i)8-s + 660. i·9-s + (581. + 581. i)10-s + (904. − 904. i)11-s + (−112. − 112. i)12-s + 1.22e3i·13-s + (−1.50e3 + 1.50e3i)14-s + (−718. − 718. i)15-s + 2.48e3·16-s + (5.50e3 − 5.50e3i)17-s + ⋯
L(s)  = 1  + (0.590 − 0.590i)2-s + (−0.216 + 0.216i)3-s + 0.301i·4-s + 0.983i·5-s + 0.255i·6-s − 0.925·7-s + (0.769 + 0.769i)8-s + 0.906i·9-s + (0.581 + 0.581i)10-s + (0.679 − 0.679i)11-s + (−0.0652 − 0.0652i)12-s + 0.555i·13-s + (−0.547 + 0.547i)14-s + (−0.212 − 0.212i)15-s + 0.607·16-s + (1.12 − 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(6.67156\)
Root analytic conductor: \(2.58293\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :3),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.48377 + 0.994885i\)
\(L(\frac12)\) \(\approx\) \(1.48377 + 0.994885i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-4.82e3 + 2.39e4i)T \)
good2 \( 1 + (-4.72 + 4.72i)T - 64iT^{2} \)
3 \( 1 + (5.84 - 5.84i)T - 729iT^{2} \)
5 \( 1 - 122. iT - 1.56e4T^{2} \)
7 \( 1 + 317.T + 1.17e5T^{2} \)
11 \( 1 + (-904. + 904. i)T - 1.77e6iT^{2} \)
13 \( 1 - 1.22e3iT - 4.82e6T^{2} \)
17 \( 1 + (-5.50e3 + 5.50e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (9.02e3 - 9.02e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 9.17e3T + 1.48e8T^{2} \)
31 \( 1 + (-2.19e4 + 2.19e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (-2.73e4 - 2.73e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (-7.54e4 - 7.54e4i)T + 4.75e9iT^{2} \)
43 \( 1 + (-4.21e4 + 4.21e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-7.80e4 - 7.80e4i)T + 1.07e10iT^{2} \)
53 \( 1 - 1.87e5T + 2.21e10T^{2} \)
59 \( 1 + 1.34e5T + 4.21e10T^{2} \)
61 \( 1 + (-2.01e5 + 2.01e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 1.56e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.55e4iT - 1.28e11T^{2} \)
73 \( 1 + (1.78e5 + 1.78e5i)T + 1.51e11iT^{2} \)
79 \( 1 + (-1.64e5 + 1.64e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 1.37e5T + 3.26e11T^{2} \)
89 \( 1 + (8.62e5 - 8.62e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-3.38e5 - 3.38e5i)T + 8.32e11iT^{2} \)
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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

−16.26945740736473010471614488075, −14.40914716080511198263894838024, −13.60181670893653304648472188091, −12.17001754067290790477574965702, −11.12798239933506591686949290347, −9.956942237052041149848249569198, −7.87287705670583232430295319177, −6.17174990186417126258139558115, −4.04713520470362734336118662904, −2.64119059985058119263082659354, 0.898239467550195250861424455126, 4.13585204979046087164348980004, 5.78727857384510135179342225306, 6.87651542582063751628313854927, 8.967690196007803259697917950476, 10.25277661417401265809025804687, 12.44524460755553202723037953679, 12.88583448677573707612791837196, 14.52696836109143485569071024699, 15.49048514778314125709319367908

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