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Properties

Label 2-29-29.17-c14-0-24
Degree $2$
Conductor $29$
Sign $0.945 + 0.325i$
Analytic cond. $36.0554$
Root an. cond. $6.00461$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−48.4 + 48.4i)2-s + (2.91e3 − 2.91e3i)3-s + 1.16e4i·4-s + 5.65e4i·5-s + 2.81e5i·6-s + 1.66e5·7-s + (−1.35e6 − 1.35e6i)8-s − 1.21e7i·9-s + (−2.73e6 − 2.73e6i)10-s + (8.97e6 − 8.97e6i)11-s + (3.40e7 + 3.40e7i)12-s + 5.10e7i·13-s + (−8.08e6 + 8.08e6i)14-s + (1.64e8 + 1.64e8i)15-s − 5.98e7·16-s + (5.50e8 − 5.50e8i)17-s + ⋯
L(s)  = 1  + (−0.378 + 0.378i)2-s + (1.33 − 1.33i)3-s + 0.713i·4-s + 0.724i·5-s + 1.00i·6-s + 0.202·7-s + (−0.648 − 0.648i)8-s − 2.54i·9-s + (−0.273 − 0.273i)10-s + (0.460 − 0.460i)11-s + (0.949 + 0.949i)12-s + 0.813i·13-s + (−0.0767 + 0.0767i)14-s + (0.963 + 0.963i)15-s − 0.222·16-s + (1.34 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(36.0554\)
Root analytic conductor: \(6.00461\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :7),\ 0.945 + 0.325i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(2.824885568\)
\(L(\frac12)\) \(\approx\) \(2.824885568\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-1.70e10 - 2.42e9i)T \)
good2 \( 1 + (48.4 - 48.4i)T - 1.63e4iT^{2} \)
3 \( 1 + (-2.91e3 + 2.91e3i)T - 4.78e6iT^{2} \)
5 \( 1 - 5.65e4iT - 6.10e9T^{2} \)
7 \( 1 - 1.66e5T + 6.78e11T^{2} \)
11 \( 1 + (-8.97e6 + 8.97e6i)T - 3.79e14iT^{2} \)
13 \( 1 - 5.10e7iT - 3.93e15T^{2} \)
17 \( 1 + (-5.50e8 + 5.50e8i)T - 1.68e17iT^{2} \)
19 \( 1 + (3.93e8 - 3.93e8i)T - 7.99e17iT^{2} \)
23 \( 1 - 2.69e9T + 1.15e19T^{2} \)
31 \( 1 + (1.63e10 - 1.63e10i)T - 7.56e20iT^{2} \)
37 \( 1 + (-1.82e10 - 1.82e10i)T + 9.01e21iT^{2} \)
41 \( 1 + (-1.65e11 - 1.65e11i)T + 3.79e22iT^{2} \)
43 \( 1 + (-3.51e11 + 3.51e11i)T - 7.38e22iT^{2} \)
47 \( 1 + (1.60e11 + 1.60e11i)T + 2.56e23iT^{2} \)
53 \( 1 - 1.48e11T + 1.37e24T^{2} \)
59 \( 1 - 2.18e12T + 6.19e24T^{2} \)
61 \( 1 + (-4.21e12 + 4.21e12i)T - 9.87e24iT^{2} \)
67 \( 1 - 1.29e12iT - 3.67e25T^{2} \)
71 \( 1 - 1.80e12iT - 8.27e25T^{2} \)
73 \( 1 + (3.74e12 + 3.74e12i)T + 1.22e26iT^{2} \)
79 \( 1 + (-6.34e12 + 6.34e12i)T - 3.68e26iT^{2} \)
83 \( 1 + 3.51e13T + 7.36e26T^{2} \)
89 \( 1 + (-1.49e13 + 1.49e13i)T - 1.95e27iT^{2} \)
97 \( 1 + (-4.05e13 - 4.05e13i)T + 6.52e27iT^{2} \)
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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

−14.04925369439882895347778881317, −12.72584953462096050973659028244, −11.71120403751271107634081278960, −9.366126489078292622360912681253, −8.387215999952286298587103209603, −7.31661202878725793697864005799, −6.63227888740903113123181278123, −3.49558423012987494804821037082, −2.60710225026581478754303957342, −0.976360755032534319974088290918, 1.21466932264806311672976234576, 2.68331402415060265004653038156, 4.20749620282045439469931650306, 5.40590863153290027960722969472, 8.124898447781048095793077776680, 9.044950750484989751381251197381, 9.961104687053840983059916047248, 10.86228995658619623138614542486, 12.85202734190196486466977334824, 14.50736201031604965221679485813

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