Properties

Label 2-29-29.20-c5-0-7
Degree $2$
Conductor $29$
Sign $-0.0428 + 0.999i$
Analytic cond. $4.65113$
Root an. cond. $2.15664$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 + 0.663i)2-s + (−4.45 − 5.58i)3-s + (−18.4 + 23.1i)4-s + (71.8 − 34.6i)5-s + (9.84 + 4.74i)6-s + (−153. − 191. i)7-s + (20.9 − 91.8i)8-s + (42.7 − 187. i)9-s + (−76.0 + 95.3i)10-s + (22.8 + 100. i)11-s + 212.·12-s + (−70.1 − 307. i)13-s + (337. + 162. i)14-s + (−513. − 247. i)15-s + (−179. − 784. i)16-s + 1.22e3·17-s + ⋯
L(s)  = 1  + (−0.243 + 0.117i)2-s + (−0.285 − 0.358i)3-s + (−0.577 + 0.724i)4-s + (1.28 − 0.619i)5-s + (0.111 + 0.0537i)6-s + (−1.18 − 1.48i)7-s + (0.115 − 0.507i)8-s + (0.175 − 0.769i)9-s + (−0.240 + 0.301i)10-s + (0.0570 + 0.249i)11-s + 0.425·12-s + (−0.115 − 0.504i)13-s + (0.460 + 0.221i)14-s + (−0.589 − 0.283i)15-s + (−0.174 − 0.766i)16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0428 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0428 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.0428 + 0.999i$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ -0.0428 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.680950 - 0.710811i\)
\(L(\frac12)\) \(\approx\) \(0.680950 - 0.710811i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (2.17e3 + 3.97e3i)T \)
good2 \( 1 + (1.37 - 0.663i)T + (19.9 - 25.0i)T^{2} \)
3 \( 1 + (4.45 + 5.58i)T + (-54.0 + 236. i)T^{2} \)
5 \( 1 + (-71.8 + 34.6i)T + (1.94e3 - 2.44e3i)T^{2} \)
7 \( 1 + (153. + 191. i)T + (-3.73e3 + 1.63e4i)T^{2} \)
11 \( 1 + (-22.8 - 100. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (70.1 + 307. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 - 1.22e3T + 1.41e6T^{2} \)
19 \( 1 + (1.32e3 - 1.66e3i)T + (-5.50e5 - 2.41e6i)T^{2} \)
23 \( 1 + (-355. - 171. i)T + (4.01e6 + 5.03e6i)T^{2} \)
31 \( 1 + (928. - 447. i)T + (1.78e7 - 2.23e7i)T^{2} \)
37 \( 1 + (-1.64e3 + 7.20e3i)T + (-6.24e7 - 3.00e7i)T^{2} \)
41 \( 1 - 8.18e3T + 1.15e8T^{2} \)
43 \( 1 + (-2.09e4 - 1.00e4i)T + (9.16e7 + 1.14e8i)T^{2} \)
47 \( 1 + (3.57e3 + 1.56e4i)T + (-2.06e8 + 9.95e7i)T^{2} \)
53 \( 1 + (9.63e3 - 4.63e3i)T + (2.60e8 - 3.26e8i)T^{2} \)
59 \( 1 - 2.82e4T + 7.14e8T^{2} \)
61 \( 1 + (7.00e3 + 8.77e3i)T + (-1.87e8 + 8.23e8i)T^{2} \)
67 \( 1 + (-4.20e3 + 1.84e4i)T + (-1.21e9 - 5.85e8i)T^{2} \)
71 \( 1 + (-1.36e4 - 5.99e4i)T + (-1.62e9 + 7.82e8i)T^{2} \)
73 \( 1 + (-2.70e4 - 1.30e4i)T + (1.29e9 + 1.62e9i)T^{2} \)
79 \( 1 + (-1.40e4 + 6.15e4i)T + (-2.77e9 - 1.33e9i)T^{2} \)
83 \( 1 + (-2.04e4 + 2.56e4i)T + (-8.76e8 - 3.84e9i)T^{2} \)
89 \( 1 + (-6.02e4 + 2.89e4i)T + (3.48e9 - 4.36e9i)T^{2} \)
97 \( 1 + (-3.17e4 + 3.98e4i)T + (-1.91e9 - 8.37e9i)T^{2} \)
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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

−16.38976005392689545782911076471, −14.16521424589391068698982108164, −12.95959405730095796379084014048, −12.64444035957647231215186410948, −10.10304781470097704904791887837, −9.406613533613007276136345872278, −7.51439406228865409407297715525, −6.07573034677351707129347349554, −3.83779775213612640601754438219, −0.72602780837050622823510042902, 2.35592505217017940311026917638, 5.33670995806594985527249978484, 6.27178561426189754984285518418, 9.066547832710319943967853630234, 9.801146985184957955754666866140, 10.89664335448975596154329693099, 12.82965896644859078838903992825, 13.96620518252244417366027692760, 15.12041547387883688572798511481, 16.44345350937727568654419311405

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