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Properties

Label 2-29-29.28-c15-0-1
Degree $2$
Conductor $29$
Sign $-0.390 + 0.920i$
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  + 169. i·2-s + 817. i·3-s + 3.90e3·4-s + 5.86e4·5-s − 1.38e5·6-s − 2.72e6·7-s + 6.23e6i·8-s + 1.36e7·9-s + 9.96e6i·10-s − 4.12e7i·11-s + 3.19e6i·12-s − 1.27e8·13-s − 4.63e8i·14-s + 4.79e7i·15-s − 9.30e8·16-s + 1.09e9i·17-s + ⋯
L(s)  = 1  + 0.938i·2-s + 0.215i·3-s + 0.119·4-s + 0.335·5-s − 0.202·6-s − 1.25·7-s + 1.05i·8-s + 0.953·9-s + 0.315i·10-s − 0.637i·11-s + 0.0257i·12-s − 0.562·13-s − 1.17i·14-s + 0.0725i·15-s − 0.866·16-s + 0.650i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.390 + 0.920i$
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: \((2,\ 29,\ (\ :15/2),\ -0.390 + 0.920i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.4083207883\)
\(L(\frac12)\) \(\approx\) \(0.4083207883\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.62e10 - 8.55e10i)T \)
good2 \( 1 - 169. iT - 3.27e4T^{2} \)
3 \( 1 - 817. iT - 1.43e7T^{2} \)
5 \( 1 - 5.86e4T + 3.05e10T^{2} \)
7 \( 1 + 2.72e6T + 4.74e12T^{2} \)
11 \( 1 + 4.12e7iT - 4.17e15T^{2} \)
13 \( 1 + 1.27e8T + 5.11e16T^{2} \)
17 \( 1 - 1.09e9iT - 2.86e18T^{2} \)
19 \( 1 - 1.07e9iT - 1.51e19T^{2} \)
23 \( 1 + 2.61e10T + 2.66e20T^{2} \)
31 \( 1 + 8.60e10iT - 2.34e22T^{2} \)
37 \( 1 + 8.70e11iT - 3.33e23T^{2} \)
41 \( 1 - 2.48e11iT - 1.55e24T^{2} \)
43 \( 1 + 1.01e12iT - 3.17e24T^{2} \)
47 \( 1 + 1.95e12iT - 1.20e25T^{2} \)
53 \( 1 + 5.78e12T + 7.31e25T^{2} \)
59 \( 1 - 1.84e13T + 3.65e26T^{2} \)
61 \( 1 + 2.21e13iT - 6.02e26T^{2} \)
67 \( 1 + 5.47e13T + 2.46e27T^{2} \)
71 \( 1 + 3.27e13T + 5.87e27T^{2} \)
73 \( 1 + 9.03e13iT - 8.90e27T^{2} \)
79 \( 1 + 3.62e12iT - 2.91e28T^{2} \)
83 \( 1 - 9.36e13T + 6.11e28T^{2} \)
89 \( 1 - 3.62e14iT - 1.74e29T^{2} \)
97 \( 1 - 1.39e15iT - 6.33e29T^{2} \)
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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

−14.77263167262153652860436170556, −13.49307975950637607525037593912, −12.25013938391420285712509385719, −10.52407524463418890637522144622, −9.420884404574274509428122364491, −7.79873442720040739558805673592, −6.57628596413418257649922269937, −5.64424936450703330109190629841, −3.76485332550176129606422067341, −2.03854177820724288674112869293, 0.10236805342126160635464023765, 1.63852429401715641256515263314, 2.76074235092084647046020230613, 4.19110728364692960543487297707, 6.30545252906668045151607765978, 7.35803966199431082908175421307, 9.794387886096071798422916419930, 9.954342300323250929457286084371, 11.78970360461099710901593835396, 12.66350215178741350344848925610

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