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Properties

Label 2-29-29.28-c15-0-31
Degree $2$
Conductor $29$
Sign $-0.882 + 0.470i$
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  − 10.1i·2-s − 4.60e3i·3-s + 3.26e4·4-s + 9.89e4·5-s − 4.65e4·6-s − 6.32e5·7-s − 6.62e5i·8-s − 6.84e6·9-s − 1.00e6i·10-s − 2.29e7i·11-s − 1.50e8i·12-s − 3.60e8·13-s + 6.39e6i·14-s − 4.55e8i·15-s + 1.06e9·16-s − 1.04e9i·17-s + ⋯
L(s)  = 1  − 0.0559i·2-s − 1.21i·3-s + 0.996·4-s + 0.566·5-s − 0.0679·6-s − 0.290·7-s − 0.111i·8-s − 0.477·9-s − 0.0316i·10-s − 0.354i·11-s − 1.21i·12-s − 1.59·13-s + 0.0162i·14-s − 0.688i·15-s + 0.990·16-s − 0.616i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.882 + 0.470i$
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: \((2,\ 29,\ (\ :15/2),\ -0.882 + 0.470i)\)

Particular Values

\(L(8)\) \(\approx\) \(2.210062534\)
\(L(\frac12)\) \(\approx\) \(2.210062534\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (8.19e10 - 4.37e10i)T \)
good2 \( 1 + 10.1iT - 3.27e4T^{2} \)
3 \( 1 + 4.60e3iT - 1.43e7T^{2} \)
5 \( 1 - 9.89e4T + 3.05e10T^{2} \)
7 \( 1 + 6.32e5T + 4.74e12T^{2} \)
11 \( 1 + 2.29e7iT - 4.17e15T^{2} \)
13 \( 1 + 3.60e8T + 5.11e16T^{2} \)
17 \( 1 + 1.04e9iT - 2.86e18T^{2} \)
19 \( 1 + 4.76e9iT - 1.51e19T^{2} \)
23 \( 1 - 2.47e10T + 2.66e20T^{2} \)
31 \( 1 - 1.81e11iT - 2.34e22T^{2} \)
37 \( 1 + 1.08e12iT - 3.33e23T^{2} \)
41 \( 1 + 1.29e12iT - 1.55e24T^{2} \)
43 \( 1 + 2.10e12iT - 3.17e24T^{2} \)
47 \( 1 - 2.66e12iT - 1.20e25T^{2} \)
53 \( 1 - 2.79e12T + 7.31e25T^{2} \)
59 \( 1 + 2.26e13T + 3.65e26T^{2} \)
61 \( 1 - 1.11e13iT - 6.02e26T^{2} \)
67 \( 1 + 1.28e13T + 2.46e27T^{2} \)
71 \( 1 - 2.16e13T + 5.87e27T^{2} \)
73 \( 1 - 5.22e13iT - 8.90e27T^{2} \)
79 \( 1 - 1.17e14iT - 2.91e28T^{2} \)
83 \( 1 - 1.31e14T + 6.11e28T^{2} \)
89 \( 1 - 7.95e14iT - 1.74e29T^{2} \)
97 \( 1 - 3.46e14iT - 6.33e29T^{2} \)
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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

−12.96818024939221156951286221614, −12.13643340087434438510357757054, −10.85610830791708739102205445696, −9.317005479751333920722898529512, −7.35811427969677862662748517418, −6.89017449603522766125090784278, −5.40893484671166123456859948069, −2.82055831833391744948362942801, −1.92534315542700295468949850666, −0.55196719416442923004032719797, 1.82264837433156204731506750302, 3.20431584791461575578012063509, 4.79244645598721169709541946857, 6.14849663873979018201873837255, 7.63917250924205037927378729585, 9.653014703115947283160724022933, 10.15878826403046033547558427739, 11.51797986449631749809588903489, 12.86987829357559023702569332306, 14.80943300668489404397096863636

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