↑

Properties

Label 2-29-29.17-c6-0-2
Degree $2$
Conductor $29$
Sign $-0.999 - 0.00470i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−11.1 + 11.1i)2-s + (25.3 − 25.3i)3-s − 184. i·4-s + 151. i·5-s + 563. i·6-s − 332.·7-s + (1.33e3 + 1.33e3i)8-s − 552. i·9-s + (−1.68e3 − 1.68e3i)10-s + (−930. + 930. i)11-s + (−4.66e3 − 4.66e3i)12-s + 565. i·13-s + (3.69e3 − 3.69e3i)14-s + (3.82e3 + 3.82e3i)15-s − 1.80e4·16-s + (−2.48e3 + 2.48e3i)17-s + ⋯
L(s)  = 1  + (−1.39 + 1.39i)2-s + (0.937 − 0.937i)3-s − 2.87i·4-s + 1.20i·5-s + 2.61i·6-s − 0.968·7-s + (2.61 + 2.61i)8-s − 0.758i·9-s + (−1.68 − 1.68i)10-s + (−0.698 + 0.698i)11-s + (−2.69 − 2.69i)12-s + 0.257i·13-s + (1.34 − 1.34i)14-s + (1.13 + 1.13i)15-s − 4.40·16-s + (−0.506 + 0.506i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00470i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 - 0.00470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.999 - 0.00470i$
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: \((2,\ 29,\ (\ :3),\ -0.999 - 0.00470i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00117658 + 0.500652i\)
\(L(\frac12)\) \(\approx\) \(0.00117658 + 0.500652i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (2.39e4 - 4.49e3i)T \)
good2 \( 1 + (11.1 - 11.1i)T - 64iT^{2} \)
3 \( 1 + (-25.3 + 25.3i)T - 729iT^{2} \)
5 \( 1 - 151. iT - 1.56e4T^{2} \)
7 \( 1 + 332.T + 1.17e5T^{2} \)
11 \( 1 + (930. - 930. i)T - 1.77e6iT^{2} \)
13 \( 1 - 565. iT - 4.82e6T^{2} \)
17 \( 1 + (2.48e3 - 2.48e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (2.51e3 - 2.51e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 1.12e4T + 1.48e8T^{2} \)
31 \( 1 + (537. - 537. i)T - 8.87e8iT^{2} \)
37 \( 1 + (-2.75e4 - 2.75e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (-8.25e4 - 8.25e4i)T + 4.75e9iT^{2} \)
43 \( 1 + (-5.73e4 + 5.73e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (1.06e5 + 1.06e5i)T + 1.07e10iT^{2} \)
53 \( 1 - 1.07e5T + 2.21e10T^{2} \)
59 \( 1 + 1.18e4T + 4.21e10T^{2} \)
61 \( 1 + (1.26e4 - 1.26e4i)T - 5.15e10iT^{2} \)
67 \( 1 + 1.44e5iT - 9.04e10T^{2} \)
71 \( 1 + 7.26e4iT - 1.28e11T^{2} \)
73 \( 1 + (-7.03e4 - 7.03e4i)T + 1.51e11iT^{2} \)
79 \( 1 + (-1.35e5 + 1.35e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 5.82e5T + 3.26e11T^{2} \)
89 \( 1 + (-6.72e4 + 6.72e4i)T - 4.96e11iT^{2} \)
97 \( 1 + (6.99e5 + 6.99e5i)T + 8.32e11iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.42475196171155991082928522734, −15.19093472055958840594926982820, −14.47618179691534373102627753656, −13.26300939729722464765816124060, −10.62604974909567077846535734338, −9.540963197499889971507892154228, −8.098704764545313634682034725864, −7.15537825062022293950098318439, −6.29190786830443712983342974030, −2.15977501928924782539724488213, 0.36775256110276887330015060913, 2.68473795689612902783758049798, 4.05514251776707654835660725468, 8.028098522088797700063160282756, 9.061296985100051337544231563190, 9.626821912639270781928382950447, 10.89952111627189873518873574969, 12.53214570123300576032385793016, 13.38338283483982045662950981246, 15.97474014929739539577260641651

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