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Properties

Label 2-29-29.28-c15-0-12
Degree $2$
Conductor $29$
Sign $-0.846 + 0.532i$
Analytic cond. $41.3811$
Root an. cond. $6.43281$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 231. i·2-s + 885. i·3-s − 2.08e4·4-s − 9.75e4·5-s − 2.05e5·6-s + 4.15e5·7-s + 2.75e6i·8-s + 1.35e7·9-s − 2.25e7i·10-s + 1.13e8i·11-s − 1.84e7i·12-s + 2.83e8·13-s + 9.62e7i·14-s − 8.63e7i·15-s − 1.32e9·16-s − 2.36e8i·17-s + ⋯
L(s)  = 1  + 1.27i·2-s + 0.233i·3-s − 0.636·4-s − 0.558·5-s − 0.299·6-s + 0.190·7-s + 0.464i·8-s + 0.945·9-s − 0.714i·10-s + 1.75i·11-s − 0.148i·12-s + 1.25·13-s + 0.243i·14-s − 0.130i·15-s − 1.23·16-s − 0.139i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.846 + 0.532i$
Analytic conductor: \(41.3811\)
Root analytic conductor: \(6.43281\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :15/2),\ -0.846 + 0.532i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.788574284\)
\(L(\frac12)\) \(\approx\) \(1.788574284\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (7.86e10 - 4.95e10i)T \)
good2 \( 1 - 231. iT - 3.27e4T^{2} \)
3 \( 1 - 885. iT - 1.43e7T^{2} \)
5 \( 1 + 9.75e4T + 3.05e10T^{2} \)
7 \( 1 - 4.15e5T + 4.74e12T^{2} \)
11 \( 1 - 1.13e8iT - 4.17e15T^{2} \)
13 \( 1 - 2.83e8T + 5.11e16T^{2} \)
17 \( 1 + 2.36e8iT - 2.86e18T^{2} \)
19 \( 1 - 1.25e9iT - 1.51e19T^{2} \)
23 \( 1 - 1.49e10T + 2.66e20T^{2} \)
31 \( 1 - 1.08e11iT - 2.34e22T^{2} \)
37 \( 1 - 6.57e11iT - 3.33e23T^{2} \)
41 \( 1 + 2.31e12iT - 1.55e24T^{2} \)
43 \( 1 + 1.23e12iT - 3.17e24T^{2} \)
47 \( 1 + 4.07e12iT - 1.20e25T^{2} \)
53 \( 1 - 1.27e13T + 7.31e25T^{2} \)
59 \( 1 + 3.52e13T + 3.65e26T^{2} \)
61 \( 1 - 4.04e13iT - 6.02e26T^{2} \)
67 \( 1 + 7.17e13T + 2.46e27T^{2} \)
71 \( 1 - 1.08e14T + 5.87e27T^{2} \)
73 \( 1 - 4.47e13iT - 8.90e27T^{2} \)
79 \( 1 + 2.34e14iT - 2.91e28T^{2} \)
83 \( 1 - 4.41e13T + 6.11e28T^{2} \)
89 \( 1 - 3.60e14iT - 1.74e29T^{2} \)
97 \( 1 - 8.83e14iT - 6.33e29T^{2} \)
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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

−15.01611137999776497695924903074, −13.46367048561840098483848106512, −12.04718039269817033385942931180, −10.52248664195229018333538380544, −8.975202435026481288278170148877, −7.58700474897166856956344946564, −6.82625609108410842553992507953, −5.16390419227052705232199946214, −3.99046538293642963550739142662, −1.68465222656199180367705734175, 0.53113473417439917231071987843, 1.47838163413804177225526092402, 3.14441398671888977557511517480, 4.12897884318351465252915879901, 6.24065732304378440119283884116, 7.924532211579169435559064319286, 9.351555056502426853153725518529, 10.97022909351693430049629088791, 11.36835607563903862410838195271, 12.86212395156391434996287259335

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