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Properties

Label 2-29-29.17-c8-0-7
Degree $2$
Conductor $29$
Sign $0.0856 - 0.996i$
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  + (8.30 − 8.30i)2-s + (−37.9 + 37.9i)3-s + 118. i·4-s − 774. i·5-s + 630. i·6-s − 18.6·7-s + (3.10e3 + 3.10e3i)8-s + 3.67e3i·9-s + (−6.43e3 − 6.43e3i)10-s + (−1.53e4 + 1.53e4i)11-s + (−4.48e3 − 4.48e3i)12-s + 4.85e4i·13-s + (−154. + 154. i)14-s + (2.94e4 + 2.94e4i)15-s + 2.13e4·16-s + (1.66e4 − 1.66e4i)17-s + ⋯
L(s)  = 1  + (0.518 − 0.518i)2-s + (−0.468 + 0.468i)3-s + 0.461i·4-s − 1.23i·5-s + 0.486i·6-s − 0.00775·7-s + (0.758 + 0.758i)8-s + 0.560i·9-s + (−0.643 − 0.643i)10-s + (−1.04 + 1.04i)11-s + (−0.216 − 0.216i)12-s + 1.70i·13-s + (−0.00402 + 0.00402i)14-s + (0.580 + 0.580i)15-s + 0.325·16-s + (0.199 − 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0856 - 0.996i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0856 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.0856 - 0.996i$
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: \((2,\ 29,\ (\ :4),\ 0.0856 - 0.996i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.08702 + 0.997624i\)
\(L(\frac12)\) \(\approx\) \(1.08702 + 0.997624i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-7.37e4 - 7.03e5i)T \)
good2 \( 1 + (-8.30 + 8.30i)T - 256iT^{2} \)
3 \( 1 + (37.9 - 37.9i)T - 6.56e3iT^{2} \)
5 \( 1 + 774. iT - 3.90e5T^{2} \)
7 \( 1 + 18.6T + 5.76e6T^{2} \)
11 \( 1 + (1.53e4 - 1.53e4i)T - 2.14e8iT^{2} \)
13 \( 1 - 4.85e4iT - 8.15e8T^{2} \)
17 \( 1 + (-1.66e4 + 1.66e4i)T - 6.97e9iT^{2} \)
19 \( 1 + (-774. + 774. i)T - 1.69e10iT^{2} \)
23 \( 1 + 5.08e3T + 7.83e10T^{2} \)
31 \( 1 + (-3.16e5 + 3.16e5i)T - 8.52e11iT^{2} \)
37 \( 1 + (2.03e6 + 2.03e6i)T + 3.51e12iT^{2} \)
41 \( 1 + (-2.10e6 - 2.10e6i)T + 7.98e12iT^{2} \)
43 \( 1 + (-3.98e6 + 3.98e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-3.40e6 - 3.40e6i)T + 2.38e13iT^{2} \)
53 \( 1 + 6.96e6T + 6.22e13T^{2} \)
59 \( 1 - 7.75e6T + 1.46e14T^{2} \)
61 \( 1 + (1.23e7 - 1.23e7i)T - 1.91e14iT^{2} \)
67 \( 1 + 1.63e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.55e7iT - 6.45e14T^{2} \)
73 \( 1 + (1.15e7 + 1.15e7i)T + 8.06e14iT^{2} \)
79 \( 1 + (-4.27e7 + 4.27e7i)T - 1.51e15iT^{2} \)
83 \( 1 - 7.14e7T + 2.25e15T^{2} \)
89 \( 1 + (5.02e7 - 5.02e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-6.30e7 - 6.30e7i)T + 7.83e15iT^{2} \)
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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

−15.98985476110267614886275487406, −13.99598033799768201958260139153, −12.84267160816339764671893162973, −12.01666706245362139409922996111, −10.73495320130607292932617116934, −9.133640028622954152761407764097, −7.60531949486254897679567511101, −5.07298013473901571737391917898, −4.36173326580531668099776095648, −2.02427907051790967577270606694, 0.57785097746924281485959991245, 3.14319064075341044778695048928, 5.54865630743305849849685238912, 6.41502272073762122943669520523, 7.76758565181613807595956265002, 10.19775604690145729039465275312, 11.02344333001849749642414849240, 12.77403601666320351283272944026, 13.89324911080349472414627735096, 15.05757230119033814006383976771

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