Properties

Label 2-29-1.1-c17-0-32
Degree $2$
Conductor $29$
Sign $1$
Analytic cond. $53.1344$
Root an. cond. $7.28933$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 616.·2-s + 1.77e4·3-s + 2.49e5·4-s + 7.68e5·5-s + 1.09e7·6-s + 1.64e6·7-s + 7.30e7·8-s + 1.86e8·9-s + 4.73e8·10-s − 6.23e8·11-s + 4.43e9·12-s + 1.86e9·13-s + 1.01e9·14-s + 1.36e10·15-s + 1.23e10·16-s − 1.09e10·17-s + 1.14e11·18-s − 3.02e10·19-s + 1.91e11·20-s + 2.92e10·21-s − 3.84e11·22-s + 2.62e11·23-s + 1.29e12·24-s − 1.72e11·25-s + 1.15e12·26-s + 1.01e12·27-s + 4.10e11·28-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.56·3-s + 1.90·4-s + 0.879·5-s + 2.66·6-s + 0.107·7-s + 1.53·8-s + 1.44·9-s + 1.49·10-s − 0.876·11-s + 2.97·12-s + 0.633·13-s + 0.183·14-s + 1.37·15-s + 0.719·16-s − 0.381·17-s + 2.45·18-s − 0.408·19-s + 1.67·20-s + 0.168·21-s − 1.49·22-s + 0.697·23-s + 2.40·24-s − 0.226·25-s + 1.08·26-s + 0.691·27-s + 0.205·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(53.1344\)
Root analytic conductor: \(7.28933\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(11.27678733\)
\(L(\frac12)\) \(\approx\) \(11.27678733\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 5.00e11T \)
good2 \( 1 - 616.T + 1.31e5T^{2} \)
3 \( 1 - 1.77e4T + 1.29e8T^{2} \)
5 \( 1 - 7.68e5T + 7.62e11T^{2} \)
7 \( 1 - 1.64e6T + 2.32e14T^{2} \)
11 \( 1 + 6.23e8T + 5.05e17T^{2} \)
13 \( 1 - 1.86e9T + 8.65e18T^{2} \)
17 \( 1 + 1.09e10T + 8.27e20T^{2} \)
19 \( 1 + 3.02e10T + 5.48e21T^{2} \)
23 \( 1 - 2.62e11T + 1.41e23T^{2} \)
31 \( 1 + 4.65e12T + 2.25e25T^{2} \)
37 \( 1 - 2.56e13T + 4.56e26T^{2} \)
41 \( 1 - 4.31e13T + 2.61e27T^{2} \)
43 \( 1 + 6.43e13T + 5.87e27T^{2} \)
47 \( 1 + 1.91e14T + 2.66e28T^{2} \)
53 \( 1 + 7.46e14T + 2.05e29T^{2} \)
59 \( 1 - 2.00e15T + 1.27e30T^{2} \)
61 \( 1 - 1.38e15T + 2.24e30T^{2} \)
67 \( 1 + 2.66e15T + 1.10e31T^{2} \)
71 \( 1 - 6.06e15T + 2.96e31T^{2} \)
73 \( 1 - 7.52e15T + 4.74e31T^{2} \)
79 \( 1 - 2.43e16T + 1.81e32T^{2} \)
83 \( 1 - 1.12e16T + 4.21e32T^{2} \)
89 \( 1 - 3.40e16T + 1.37e33T^{2} \)
97 \( 1 + 7.90e16T + 5.95e33T^{2} \)
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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

−13.30438731786527490243947395924, −12.98182106214097063355230879032, −11.03661988031211978051831602257, −9.467948942870368645136409409811, −8.019203804652542460523190248049, −6.48319162805095510360627044293, −5.11214625270371993725110188695, −3.74134274539878243731220057391, −2.68466653699349489665688424772, −1.86379463366811195350989815055, 1.86379463366811195350989815055, 2.68466653699349489665688424772, 3.74134274539878243731220057391, 5.11214625270371993725110188695, 6.48319162805095510360627044293, 8.019203804652542460523190248049, 9.467948942870368645136409409811, 11.03661988031211978051831602257, 12.98182106214097063355230879032, 13.30438731786527490243947395924

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