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Properties

Label 2-29-29.28-c5-0-5
Degree $2$
Conductor $29$
Sign $0.760 - 0.649i$
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  + 3.61i·2-s − 5.13i·3-s + 18.9·4-s + 15.9·5-s + 18.5·6-s + 69.7·7-s + 184. i·8-s + 216.·9-s + 57.5i·10-s + 400. i·11-s − 97.0i·12-s − 178.·13-s + 252. i·14-s − 81.5i·15-s − 60.6·16-s − 1.41e3i·17-s + ⋯
L(s)  = 1  + 0.639i·2-s − 0.329i·3-s + 0.591·4-s + 0.284·5-s + 0.210·6-s + 0.538·7-s + 1.01i·8-s + 0.891·9-s + 0.181i·10-s + 0.997i·11-s − 0.194i·12-s − 0.293·13-s + 0.344i·14-s − 0.0936i·15-s − 0.0592·16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ 0.760 - 0.649i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.80095 + 0.664918i\)
\(L(\frac12)\) \(\approx\) \(1.80095 + 0.664918i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.44e3 - 2.94e3i)T \)
good2 \( 1 - 3.61iT - 32T^{2} \)
3 \( 1 + 5.13iT - 243T^{2} \)
5 \( 1 - 15.9T + 3.12e3T^{2} \)
7 \( 1 - 69.7T + 1.68e4T^{2} \)
11 \( 1 - 400. iT - 1.61e5T^{2} \)
13 \( 1 + 178.T + 3.71e5T^{2} \)
17 \( 1 + 1.41e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.72e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.10e3T + 6.43e6T^{2} \)
31 \( 1 - 2.03e3iT - 2.86e7T^{2} \)
37 \( 1 + 7.21e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.33e3iT - 1.15e8T^{2} \)
43 \( 1 + 4.97e3iT - 1.47e8T^{2} \)
47 \( 1 + 493. iT - 2.29e8T^{2} \)
53 \( 1 + 1.08e4T + 4.18e8T^{2} \)
59 \( 1 + 4.33e3T + 7.14e8T^{2} \)
61 \( 1 + 4.64e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.48e3T + 1.35e9T^{2} \)
71 \( 1 - 5.86e4T + 1.80e9T^{2} \)
73 \( 1 - 1.53e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.38e4iT - 3.07e9T^{2} \)
83 \( 1 - 4.57e4T + 3.93e9T^{2} \)
89 \( 1 - 1.04e5iT - 5.58e9T^{2} \)
97 \( 1 + 1.51e5iT - 8.58e9T^{2} \)
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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

−16.05752682243793045041666864101, −15.16885189185598158922973625917, −13.96334452479108232296946524339, −12.48762414887791882846624483321, −11.21799153574783177117933966867, −9.619919298339645670098322552560, −7.68433371549105659157784675757, −6.82145562236737283616336173684, −4.99051303170429702386281609336, −2.01808469993381526239639544879, 1.70327997223238695050924854688, 3.83423422322461450363135384178, 6.05927085244923403208520314160, 7.88420292805957214464905222434, 9.821045478389001959676100129279, 10.77700332583602632917282615318, 11.98874249901935746188900341250, 13.26421225367381465089690925197, 14.81615116804611109660908560966, 15.94140892648176556511101029987

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