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Properties

Label 2-29-29.28-c15-0-29
Degree $2$
Conductor $29$
Sign $-0.333 + 0.942i$
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  − 195. i·2-s + 7.18e3i·3-s − 5.35e3·4-s + 2.74e5·5-s + 1.40e6·6-s − 9.64e5·7-s − 5.35e6i·8-s − 3.72e7·9-s − 5.36e7i·10-s − 1.01e8i·11-s − 3.84e7i·12-s − 1.35e8·13-s + 1.88e8i·14-s + 1.97e9i·15-s − 1.22e9·16-s − 3.00e9i·17-s + ⋯
L(s)  = 1  − 1.07i·2-s + 1.89i·3-s − 0.163·4-s + 1.57·5-s + 2.04·6-s − 0.442·7-s − 0.902i·8-s − 2.59·9-s − 1.69i·10-s − 1.56i·11-s − 0.309i·12-s − 0.597·13-s + 0.477i·14-s + 2.97i·15-s − 1.13·16-s − 1.77i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.333 + 0.942i$
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: \((2,\ 29,\ (\ :15/2),\ -0.333 + 0.942i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.791628989\)
\(L(\frac12)\) \(\approx\) \(1.791628989\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.09e10 - 8.75e10i)T \)
good2 \( 1 + 195. iT - 3.27e4T^{2} \)
3 \( 1 - 7.18e3iT - 1.43e7T^{2} \)
5 \( 1 - 2.74e5T + 3.05e10T^{2} \)
7 \( 1 + 9.64e5T + 4.74e12T^{2} \)
11 \( 1 + 1.01e8iT - 4.17e15T^{2} \)
13 \( 1 + 1.35e8T + 5.11e16T^{2} \)
17 \( 1 + 3.00e9iT - 2.86e18T^{2} \)
19 \( 1 + 3.23e9iT - 1.51e19T^{2} \)
23 \( 1 + 1.37e10T + 2.66e20T^{2} \)
31 \( 1 + 9.55e10iT - 2.34e22T^{2} \)
37 \( 1 + 3.13e11iT - 3.33e23T^{2} \)
41 \( 1 - 1.55e11iT - 1.55e24T^{2} \)
43 \( 1 - 2.62e12iT - 3.17e24T^{2} \)
47 \( 1 - 2.17e12iT - 1.20e25T^{2} \)
53 \( 1 + 9.74e11T + 7.31e25T^{2} \)
59 \( 1 - 2.03e13T + 3.65e26T^{2} \)
61 \( 1 + 7.96e12iT - 6.02e26T^{2} \)
67 \( 1 + 3.02e13T + 2.46e27T^{2} \)
71 \( 1 - 2.59e13T + 5.87e27T^{2} \)
73 \( 1 + 1.41e14iT - 8.90e27T^{2} \)
79 \( 1 + 8.83e13iT - 2.91e28T^{2} \)
83 \( 1 - 2.22e14T + 6.11e28T^{2} \)
89 \( 1 - 2.42e14iT - 1.74e29T^{2} \)
97 \( 1 - 7.97e13iT - 6.33e29T^{2} \)
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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

−13.40939823059903525662099616480, −11.50290632471700501616486229096, −10.65977460298720929922502181128, −9.613405702603479413628698424584, −9.242364404025351985668601667658, −6.11750546817021647215746757529, −4.93255330918634570119165168908, −3.27337990759276659250304844082, −2.54549973781415914920930016742, −0.43786120819697900548781243830, 1.77826624572118796032229971943, 2.15247782760195171013800248673, 5.60426483723552869488720435585, 6.34145310212036541670687120763, 7.18266681100471378395025925635, 8.358560831391707174832841009334, 10.05265270690245651666091465457, 12.17297271592766796325451987156, 13.00426860945001000342238095088, 14.06275986151487671982606378887

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