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Properties

Label 2-29-29.17-c10-0-6
Degree $2$
Conductor $29$
Sign $0.430 - 0.902i$
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  + (29.1 − 29.1i)2-s + (−96.9 + 96.9i)3-s − 675. i·4-s − 1.08e3i·5-s + 5.65e3i·6-s − 1.76e4·7-s + (1.01e4 + 1.01e4i)8-s + 4.02e4i·9-s + (−3.17e4 − 3.17e4i)10-s + (6.61e4 − 6.61e4i)11-s + (6.55e4 + 6.55e4i)12-s + 4.40e5i·13-s + (−5.15e5 + 5.15e5i)14-s + (1.05e5 + 1.05e5i)15-s + 1.28e6·16-s + (−1.42e6 + 1.42e6i)17-s + ⋯
L(s)  = 1  + (0.910 − 0.910i)2-s + (−0.399 + 0.399i)3-s − 0.659i·4-s − 0.348i·5-s + 0.727i·6-s − 1.05·7-s + (0.309 + 0.309i)8-s + 0.681i·9-s + (−0.317 − 0.317i)10-s + (0.410 − 0.410i)11-s + (0.263 + 0.263i)12-s + 1.18i·13-s + (−0.957 + 0.957i)14-s + (0.139 + 0.139i)15-s + 1.22·16-s + (−1.00 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.430 - 0.902i$
Analytic conductor: \(18.4253\)
Root analytic conductor: \(4.29247\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5),\ 0.430 - 0.902i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.42998 + 0.902746i\)
\(L(\frac12)\) \(\approx\) \(1.42998 + 0.902746i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-5.16e6 + 1.98e7i)T \)
good2 \( 1 + (-29.1 + 29.1i)T - 1.02e3iT^{2} \)
3 \( 1 + (96.9 - 96.9i)T - 5.90e4iT^{2} \)
5 \( 1 + 1.08e3iT - 9.76e6T^{2} \)
7 \( 1 + 1.76e4T + 2.82e8T^{2} \)
11 \( 1 + (-6.61e4 + 6.61e4i)T - 2.59e10iT^{2} \)
13 \( 1 - 4.40e5iT - 1.37e11T^{2} \)
17 \( 1 + (1.42e6 - 1.42e6i)T - 2.01e12iT^{2} \)
19 \( 1 + (3.09e6 - 3.09e6i)T - 6.13e12iT^{2} \)
23 \( 1 - 1.03e7T + 4.14e13T^{2} \)
31 \( 1 + (2.89e7 - 2.89e7i)T - 8.19e14iT^{2} \)
37 \( 1 + (2.36e7 + 2.36e7i)T + 4.80e15iT^{2} \)
41 \( 1 + (4.75e6 + 4.75e6i)T + 1.34e16iT^{2} \)
43 \( 1 + (1.29e8 - 1.29e8i)T - 2.16e16iT^{2} \)
47 \( 1 + (2.61e8 + 2.61e8i)T + 5.25e16iT^{2} \)
53 \( 1 + 4.69e7T + 1.74e17T^{2} \)
59 \( 1 - 6.63e8T + 5.11e17T^{2} \)
61 \( 1 + (7.34e8 - 7.34e8i)T - 7.13e17iT^{2} \)
67 \( 1 - 9.94e7iT - 1.82e18T^{2} \)
71 \( 1 + 1.81e9iT - 3.25e18T^{2} \)
73 \( 1 + (-1.96e9 - 1.96e9i)T + 4.29e18iT^{2} \)
79 \( 1 + (-6.17e8 + 6.17e8i)T - 9.46e18iT^{2} \)
83 \( 1 - 4.18e9T + 1.55e19T^{2} \)
89 \( 1 + (-7.05e9 + 7.05e9i)T - 3.11e19iT^{2} \)
97 \( 1 + (-1.10e9 - 1.10e9i)T + 7.37e19iT^{2} \)
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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

−14.73788164732509520834975622361, −13.36931808062354894549849650841, −12.63559701866984280443237670192, −11.31321124962625774904800487942, −10.37809936025655012161790564170, −8.709820215691365540979789975445, −6.40964199699917048483638345106, −4.77761692021491818467718491670, −3.64745290549731345820339857969, −1.88705027966621263302289128218, 0.48089047363145323493877288896, 3.20383161708784939084395047343, 4.97931922651279942777713147364, 6.57220790582094503345677121127, 6.93617623785304749748006601471, 9.248511904648557925252842244162, 10.91205291236515353414429856436, 12.72343043551194829340535683999, 13.17578866672107319331639224564, 14.86632085329185925459336428148

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