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Properties

Label 2-29-29.28-c15-0-16
Degree $2$
Conductor $29$
Sign $0.190 - 0.981i$
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  + 317. i·2-s + 3.42e3i·3-s − 6.81e4·4-s − 1.76e5·5-s − 1.08e6·6-s + 9.84e4·7-s − 1.12e7i·8-s + 2.63e6·9-s − 5.59e7i·10-s − 4.71e7i·11-s − 2.33e8i·12-s + 3.71e7·13-s + 3.12e7i·14-s − 6.02e8i·15-s + 1.33e9·16-s + 1.54e7i·17-s + ⋯
L(s)  = 1  + 1.75i·2-s + 0.903i·3-s − 2.08·4-s − 1.00·5-s − 1.58·6-s + 0.0451·7-s − 1.89i·8-s + 0.183·9-s − 1.76i·10-s − 0.729i·11-s − 1.87i·12-s + 0.164·13-s + 0.0793i·14-s − 0.910i·15-s + 1.24·16-s + 0.00915i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.190 - 0.981i$
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: \((2,\ 29,\ (\ :15/2),\ 0.190 - 0.981i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.8615587943\)
\(L(\frac12)\) \(\approx\) \(0.8615587943\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-1.77e10 + 9.11e10i)T \)
good2 \( 1 - 317. iT - 3.27e4T^{2} \)
3 \( 1 - 3.42e3iT - 1.43e7T^{2} \)
5 \( 1 + 1.76e5T + 3.05e10T^{2} \)
7 \( 1 - 9.84e4T + 4.74e12T^{2} \)
11 \( 1 + 4.71e7iT - 4.17e15T^{2} \)
13 \( 1 - 3.71e7T + 5.11e16T^{2} \)
17 \( 1 - 1.54e7iT - 2.86e18T^{2} \)
19 \( 1 - 2.06e8iT - 1.51e19T^{2} \)
23 \( 1 + 9.21e8T + 2.66e20T^{2} \)
31 \( 1 + 1.66e11iT - 2.34e22T^{2} \)
37 \( 1 + 4.77e11iT - 3.33e23T^{2} \)
41 \( 1 - 1.44e12iT - 1.55e24T^{2} \)
43 \( 1 - 5.01e11iT - 3.17e24T^{2} \)
47 \( 1 + 1.87e12iT - 1.20e25T^{2} \)
53 \( 1 + 4.37e12T + 7.31e25T^{2} \)
59 \( 1 - 2.13e13T + 3.65e26T^{2} \)
61 \( 1 - 1.53e13iT - 6.02e26T^{2} \)
67 \( 1 + 2.28e13T + 2.46e27T^{2} \)
71 \( 1 - 9.05e13T + 5.87e27T^{2} \)
73 \( 1 + 5.94e13iT - 8.90e27T^{2} \)
79 \( 1 + 1.36e14iT - 2.91e28T^{2} \)
83 \( 1 + 3.25e14T + 6.11e28T^{2} \)
89 \( 1 + 1.75e14iT - 1.74e29T^{2} \)
97 \( 1 - 6.73e14iT - 6.33e29T^{2} \)
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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

−14.55798144466742849862880711418, −13.21285190586417719731699753895, −11.35980650142169303508005253130, −9.716650021628893600244158978587, −8.445902759703047303894788723393, −7.49130163966665291662549996843, −6.03579891802782085947166314817, −4.65595640700329368696049149211, −3.73699169407604337910338338984, −0.32425652885761437425533076082, 0.946730081339139393946061074848, 1.98612772473604142105520697644, 3.43933863796171047687707671828, 4.65172282933290465884172269488, 7.08977104813420178475251150525, 8.444058344614904774952598507413, 9.963953992780762191792578786511, 11.21533676605697243027580049717, 12.23010994127463197025508758535, 12.78296078298454894992681660233

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