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Properties

Label 2-29-29.17-c8-0-11
Degree $2$
Conductor $29$
Sign $0.999 - 0.00769i$
Analytic cond. $11.8139$
Root an. cond. $3.43714$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.93 − 9.93i)2-s + (27.3 − 27.3i)3-s + 58.4i·4-s + 447. i·5-s − 543. i·6-s + 2.07e3·7-s + (3.12e3 + 3.12e3i)8-s + 5.06e3i·9-s + (4.44e3 + 4.44e3i)10-s + (5.20e3 − 5.20e3i)11-s + (1.59e3 + 1.59e3i)12-s − 1.39e4i·13-s + (2.06e4 − 2.06e4i)14-s + (1.22e4 + 1.22e4i)15-s + 4.71e4·16-s + (621. − 621. i)17-s + ⋯
L(s)  = 1  + (0.621 − 0.621i)2-s + (0.337 − 0.337i)3-s + 0.228i·4-s + 0.716i·5-s − 0.419i·6-s + 0.865·7-s + (0.762 + 0.762i)8-s + 0.772i·9-s + (0.444 + 0.444i)10-s + (0.355 − 0.355i)11-s + (0.0770 + 0.0770i)12-s − 0.488i·13-s + (0.537 − 0.537i)14-s + (0.241 + 0.241i)15-s + 0.719·16-s + (0.00743 − 0.00743i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00769i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 - 0.00769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.999 - 0.00769i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ 0.999 - 0.00769i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.02945 + 0.0116502i\)
\(L(\frac12)\) \(\approx\) \(3.02945 + 0.0116502i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (6.93e5 - 1.39e5i)T \)
good2 \( 1 + (-9.93 + 9.93i)T - 256iT^{2} \)
3 \( 1 + (-27.3 + 27.3i)T - 6.56e3iT^{2} \)
5 \( 1 - 447. iT - 3.90e5T^{2} \)
7 \( 1 - 2.07e3T + 5.76e6T^{2} \)
11 \( 1 + (-5.20e3 + 5.20e3i)T - 2.14e8iT^{2} \)
13 \( 1 + 1.39e4iT - 8.15e8T^{2} \)
17 \( 1 + (-621. + 621. i)T - 6.97e9iT^{2} \)
19 \( 1 + (-2.14e4 + 2.14e4i)T - 1.69e10iT^{2} \)
23 \( 1 - 1.10e5T + 7.83e10T^{2} \)
31 \( 1 + (674. - 674. i)T - 8.52e11iT^{2} \)
37 \( 1 + (1.23e6 + 1.23e6i)T + 3.51e12iT^{2} \)
41 \( 1 + (1.64e6 + 1.64e6i)T + 7.98e12iT^{2} \)
43 \( 1 + (2.44e6 - 2.44e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (1.34e6 + 1.34e6i)T + 2.38e13iT^{2} \)
53 \( 1 - 1.63e6T + 6.22e13T^{2} \)
59 \( 1 - 1.86e7T + 1.46e14T^{2} \)
61 \( 1 + (-1.12e7 + 1.12e7i)T - 1.91e14iT^{2} \)
67 \( 1 + 1.90e7iT - 4.06e14T^{2} \)
71 \( 1 + 4.15e7iT - 6.45e14T^{2} \)
73 \( 1 + (2.37e7 + 2.37e7i)T + 8.06e14iT^{2} \)
79 \( 1 + (2.06e7 - 2.06e7i)T - 1.51e15iT^{2} \)
83 \( 1 + 1.71e7T + 2.25e15T^{2} \)
89 \( 1 + (-5.27e7 + 5.27e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (4.78e7 + 4.78e7i)T + 7.83e15iT^{2} \)
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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

−14.78855424339834200316896495080, −13.92684563122493607716125061152, −12.90892702590620106854148425641, −11.47495942449384888023352551044, −10.67801475987948470244262773562, −8.426629299795457512855272820776, −7.30046449475968927695313381134, −5.05139960376905995233420919757, −3.28676340927968478713391553031, −1.90638405684661504494731694968, 1.29412578874645745140455984435, 4.09034050395085376587134279640, 5.24250008880720799352349791135, 6.88707127081740579654628023071, 8.647435389463089193082684042284, 9.913656277484119205806101467198, 11.67201399440074548538042576884, 13.09835699353231241423213239006, 14.42918556292398506526073901164, 15.01589007261598026268655927765

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