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Properties

Label 2-29-29.12-c8-0-9
Degree $2$
Conductor $29$
Sign $-0.653 - 0.757i$
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  + (19.3 + 19.3i)2-s + (104. + 104. i)3-s + 489. i·4-s − 703. i·5-s + 4.05e3i·6-s + 1.72e3·7-s + (−4.51e3 + 4.51e3i)8-s + 1.54e4i·9-s + (1.35e4 − 1.35e4i)10-s + (−1.04e4 − 1.04e4i)11-s + (−5.13e4 + 5.13e4i)12-s − 4.59e4i·13-s + (3.32e4 + 3.32e4i)14-s + (7.38e4 − 7.38e4i)15-s − 4.88e4·16-s + (2.17e4 + 2.17e4i)17-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)2-s + (1.29 + 1.29i)3-s + 1.91i·4-s − 1.12i·5-s + 3.12i·6-s + 0.716·7-s + (−1.10 + 1.10i)8-s + 2.35i·9-s + (1.35 − 1.35i)10-s + (−0.710 − 0.710i)11-s + (−2.47 + 2.47i)12-s − 1.60i·13-s + (0.864 + 0.864i)14-s + (1.45 − 1.45i)15-s − 0.745·16-s + (0.260 + 0.260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.653 - 0.757i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ -0.653 - 0.757i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.02061 + 4.41116i\)
\(L(\frac12)\) \(\approx\) \(2.02061 + 4.41116i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-3.52e5 - 6.13e5i)T \)
good2 \( 1 + (-19.3 - 19.3i)T + 256iT^{2} \)
3 \( 1 + (-104. - 104. i)T + 6.56e3iT^{2} \)
5 \( 1 + 703. iT - 3.90e5T^{2} \)
7 \( 1 - 1.72e3T + 5.76e6T^{2} \)
11 \( 1 + (1.04e4 + 1.04e4i)T + 2.14e8iT^{2} \)
13 \( 1 + 4.59e4iT - 8.15e8T^{2} \)
17 \( 1 + (-2.17e4 - 2.17e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (1.00e5 + 1.00e5i)T + 1.69e10iT^{2} \)
23 \( 1 + 3.71e5T + 7.83e10T^{2} \)
31 \( 1 + (-7.42e5 - 7.42e5i)T + 8.52e11iT^{2} \)
37 \( 1 + (1.38e6 - 1.38e6i)T - 3.51e12iT^{2} \)
41 \( 1 + (-1.11e6 + 1.11e6i)T - 7.98e12iT^{2} \)
43 \( 1 + (3.08e5 + 3.08e5i)T + 1.16e13iT^{2} \)
47 \( 1 + (2.34e6 - 2.34e6i)T - 2.38e13iT^{2} \)
53 \( 1 + 2.43e6T + 6.22e13T^{2} \)
59 \( 1 - 1.50e7T + 1.46e14T^{2} \)
61 \( 1 + (5.29e6 + 5.29e6i)T + 1.91e14iT^{2} \)
67 \( 1 + 3.07e5iT - 4.06e14T^{2} \)
71 \( 1 - 3.43e7iT - 6.45e14T^{2} \)
73 \( 1 + (-2.97e7 + 2.97e7i)T - 8.06e14iT^{2} \)
79 \( 1 + (2.34e7 + 2.34e7i)T + 1.51e15iT^{2} \)
83 \( 1 - 3.11e7T + 2.25e15T^{2} \)
89 \( 1 + (7.74e7 + 7.74e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (-1.09e7 + 1.09e7i)T - 7.83e15iT^{2} \)
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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

−15.65440016220502287823123633785, −14.65716245216820015196292201295, −13.70377311637875277236397335907, −12.73464038592195546358609504520, −10.41610327722286563183881740706, −8.439597496085380958367923009895, −8.129758047537648532269426069313, −5.34952156472279663328737324364, −4.58919423987955021044954533318, −3.15735936183504121973523270980, 1.84141228338747421510878254142, 2.47265537247901460993328679731, 4.03207595514123705358995029637, 6.51052928002869083227738016535, 7.935533901505742350058172389864, 9.952480966462645391172754427980, 11.52943701279652615558146361218, 12.45376218984193406105842329615, 13.75136541139509742143117979099, 14.27284901248661778850867596875

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