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Properties

Label 2-29-29.28-c15-0-6
Degree $2$
Conductor $29$
Sign $0.699 + 0.714i$
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  − 255. i·2-s + 4.33e3i·3-s − 3.25e4·4-s − 2.84e5·5-s + 1.10e6·6-s − 3.50e6·7-s − 4.42e4i·8-s − 4.47e6·9-s + 7.27e7i·10-s − 3.04e7i·11-s − 1.41e8i·12-s − 4.29e8·13-s + 8.95e8i·14-s − 1.23e9i·15-s − 1.07e9·16-s + 2.53e9i·17-s + ⋯
L(s)  = 1  − 1.41i·2-s + 1.14i·3-s − 0.994·4-s − 1.62·5-s + 1.61·6-s − 1.60·7-s − 0.00746i·8-s − 0.311·9-s + 2.30i·10-s − 0.471i·11-s − 1.13i·12-s − 1.89·13-s + 2.27i·14-s − 1.86i·15-s − 1.00·16-s + 1.49i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.699 + 0.714i$
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: \((2,\ 29,\ (\ :15/2),\ 0.699 + 0.714i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.4119995586\)
\(L(\frac12)\) \(\approx\) \(0.4119995586\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-6.49e10 - 6.63e10i)T \)
good2 \( 1 + 255. iT - 3.27e4T^{2} \)
3 \( 1 - 4.33e3iT - 1.43e7T^{2} \)
5 \( 1 + 2.84e5T + 3.05e10T^{2} \)
7 \( 1 + 3.50e6T + 4.74e12T^{2} \)
11 \( 1 + 3.04e7iT - 4.17e15T^{2} \)
13 \( 1 + 4.29e8T + 5.11e16T^{2} \)
17 \( 1 - 2.53e9iT - 2.86e18T^{2} \)
19 \( 1 + 4.65e9iT - 1.51e19T^{2} \)
23 \( 1 - 5.54e9T + 2.66e20T^{2} \)
31 \( 1 + 8.17e10iT - 2.34e22T^{2} \)
37 \( 1 + 3.34e11iT - 3.33e23T^{2} \)
41 \( 1 + 4.52e11iT - 1.55e24T^{2} \)
43 \( 1 - 6.62e11iT - 3.17e24T^{2} \)
47 \( 1 + 4.60e12iT - 1.20e25T^{2} \)
53 \( 1 + 2.69e12T + 7.31e25T^{2} \)
59 \( 1 + 3.38e13T + 3.65e26T^{2} \)
61 \( 1 - 2.05e13iT - 6.02e26T^{2} \)
67 \( 1 + 3.33e13T + 2.46e27T^{2} \)
71 \( 1 - 8.83e13T + 5.87e27T^{2} \)
73 \( 1 - 1.19e14iT - 8.90e27T^{2} \)
79 \( 1 + 2.60e14iT - 2.91e28T^{2} \)
83 \( 1 + 1.84e14T + 6.11e28T^{2} \)
89 \( 1 - 8.68e13iT - 1.74e29T^{2} \)
97 \( 1 - 7.72e14iT - 6.33e29T^{2} \)
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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

−12.88288162371258364506662971311, −12.15908720451261590442178992243, −10.91349160226794468238254923767, −10.05146878995463500096399117428, −9.020522973471237755563539848149, −7.05159977817267473975896202158, −4.56797981110473937971390144088, −3.65016150213017012458924564565, −2.86316662872047693386848636598, −0.37471563547808913291631072207, 0.33320150213488374740374281051, 2.82379781368287916417186568147, 4.67491286718607666511075145586, 6.52441082824167672098497900662, 7.28664085458794507405562565208, 7.84653937165356323892592391150, 9.612447960852015524637423841162, 11.98643621486817709280621123288, 12.55700141063997662954782685436, 14.06184323364237006440469164739

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