Properties

Label 2-29-29.16-c5-0-7
Degree $2$
Conductor $29$
Sign $0.974 + 0.224i$
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  + (3.98 + 1.91i)2-s + (6.52 − 8.18i)3-s + (−7.76 − 9.73i)4-s + (64.0 + 30.8i)5-s + (41.6 − 20.0i)6-s + (85.0 − 106. i)7-s + (−43.7 − 191. i)8-s + (29.6 + 130. i)9-s + (196. + 245. i)10-s + (−58.7 + 257. i)11-s − 130.·12-s + (2.84 − 12.4i)13-s + (543. − 261. i)14-s + (670. − 323. i)15-s + (104. − 458. i)16-s − 2.14e3·17-s + ⋯
L(s)  = 1  + (0.704 + 0.339i)2-s + (0.418 − 0.525i)3-s + (−0.242 − 0.304i)4-s + (1.14 + 0.552i)5-s + (0.472 − 0.227i)6-s + (0.656 − 0.822i)7-s + (−0.241 − 1.05i)8-s + (0.122 + 0.535i)9-s + (0.620 + 0.777i)10-s + (−0.146 + 0.641i)11-s − 0.261·12-s + (0.00467 − 0.0204i)13-s + (0.740 − 0.356i)14-s + (0.769 − 0.370i)15-s + (0.102 − 0.447i)16-s − 1.80·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.974 + 0.224i$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ 0.974 + 0.224i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.52469 - 0.286535i\)
\(L(\frac12)\) \(\approx\) \(2.52469 - 0.286535i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.25e3 + 3.15e3i)T \)
good2 \( 1 + (-3.98 - 1.91i)T + (19.9 + 25.0i)T^{2} \)
3 \( 1 + (-6.52 + 8.18i)T + (-54.0 - 236. i)T^{2} \)
5 \( 1 + (-64.0 - 30.8i)T + (1.94e3 + 2.44e3i)T^{2} \)
7 \( 1 + (-85.0 + 106. i)T + (-3.73e3 - 1.63e4i)T^{2} \)
11 \( 1 + (58.7 - 257. i)T + (-1.45e5 - 6.98e4i)T^{2} \)
13 \( 1 + (-2.84 + 12.4i)T + (-3.34e5 - 1.61e5i)T^{2} \)
17 \( 1 + 2.14e3T + 1.41e6T^{2} \)
19 \( 1 + (-721. - 904. i)T + (-5.50e5 + 2.41e6i)T^{2} \)
23 \( 1 + (2.67e3 - 1.28e3i)T + (4.01e6 - 5.03e6i)T^{2} \)
31 \( 1 + (-2.73e3 - 1.31e3i)T + (1.78e7 + 2.23e7i)T^{2} \)
37 \( 1 + (-755. - 3.30e3i)T + (-6.24e7 + 3.00e7i)T^{2} \)
41 \( 1 - 1.70e4T + 1.15e8T^{2} \)
43 \( 1 + (-1.22e4 + 5.88e3i)T + (9.16e7 - 1.14e8i)T^{2} \)
47 \( 1 + (2.58e3 - 1.13e4i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (2.92e4 + 1.40e4i)T + (2.60e8 + 3.26e8i)T^{2} \)
59 \( 1 + 1.24e4T + 7.14e8T^{2} \)
61 \( 1 + (-1.48e4 + 1.86e4i)T + (-1.87e8 - 8.23e8i)T^{2} \)
67 \( 1 + (1.06e4 + 4.66e4i)T + (-1.21e9 + 5.85e8i)T^{2} \)
71 \( 1 + (-6.51e3 + 2.85e4i)T + (-1.62e9 - 7.82e8i)T^{2} \)
73 \( 1 + (9.95e3 - 4.79e3i)T + (1.29e9 - 1.62e9i)T^{2} \)
79 \( 1 + (1.06e4 + 4.68e4i)T + (-2.77e9 + 1.33e9i)T^{2} \)
83 \( 1 + (-1.29e4 - 1.62e4i)T + (-8.76e8 + 3.84e9i)T^{2} \)
89 \( 1 + (2.15e4 + 1.03e4i)T + (3.48e9 + 4.36e9i)T^{2} \)
97 \( 1 + (4.86e4 + 6.09e4i)T + (-1.91e9 + 8.37e9i)T^{2} \)
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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

−15.68906829644647047387558245170, −14.26173007137512803471231313662, −13.84420556319041047307828735301, −12.95167765230583378892490285237, −10.76686592337706573885388913260, −9.611158010254290585310320065368, −7.57099274078751105200262735688, −6.21067262106021639959779212026, −4.55107670851495836050701355135, −1.91960013735546802883531903579, 2.44645208008288585094631743902, 4.42571852926134415855700702029, 5.79416370823868261614993631829, 8.605054270949680301418472021931, 9.307725123947595408942806589478, 11.23962575502647920726005907258, 12.61302387602478295106053876530, 13.63181005333541925899259667349, 14.60003114470193535228447918492, 15.91584179876402403737295863172

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