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Properties

Label 2-29-29.28-c5-0-0
Degree $2$
Conductor $29$
Sign $-0.732 + 0.680i$
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  + 8.60i·2-s − 4.65i·3-s − 42.0·4-s − 58.8·5-s + 40.0·6-s − 192.·7-s − 86.0i·8-s + 221.·9-s − 506. i·10-s + 34.8i·11-s + 195. i·12-s − 149.·13-s − 1.65e3i·14-s + 273. i·15-s − 603.·16-s + 2.02e3i·17-s + ⋯
L(s)  = 1  + 1.52i·2-s − 0.298i·3-s − 1.31·4-s − 1.05·5-s + 0.453·6-s − 1.48·7-s − 0.475i·8-s + 0.911·9-s − 1.60i·10-s + 0.0867i·11-s + 0.391i·12-s − 0.245·13-s − 2.26i·14-s + 0.314i·15-s − 0.589·16-s + 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.732 + 0.680i$
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: \((2,\ 29,\ (\ :5/2),\ -0.732 + 0.680i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.190600 - 0.485060i\)
\(L(\frac12)\) \(\approx\) \(0.190600 - 0.485060i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-3.31e3 + 3.08e3i)T \)
good2 \( 1 - 8.60iT - 32T^{2} \)
3 \( 1 + 4.65iT - 243T^{2} \)
5 \( 1 + 58.8T + 3.12e3T^{2} \)
7 \( 1 + 192.T + 1.68e4T^{2} \)
11 \( 1 - 34.8iT - 1.61e5T^{2} \)
13 \( 1 + 149.T + 3.71e5T^{2} \)
17 \( 1 - 2.02e3iT - 1.41e6T^{2} \)
19 \( 1 - 500. iT - 2.47e6T^{2} \)
23 \( 1 + 1.36e3T + 6.43e6T^{2} \)
31 \( 1 - 6.70e3iT - 2.86e7T^{2} \)
37 \( 1 + 2.78e3iT - 6.93e7T^{2} \)
41 \( 1 + 3.74e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.79e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.90e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.07e3T + 4.18e8T^{2} \)
59 \( 1 - 3.03e4T + 7.14e8T^{2} \)
61 \( 1 - 8.37e3iT - 8.44e8T^{2} \)
67 \( 1 + 4.98e4T + 1.35e9T^{2} \)
71 \( 1 + 4.08e4T + 1.80e9T^{2} \)
73 \( 1 - 8.34e4iT - 2.07e9T^{2} \)
79 \( 1 + 9.97e4iT - 3.07e9T^{2} \)
83 \( 1 + 9.11e4T + 3.93e9T^{2} \)
89 \( 1 - 1.15e5iT - 5.58e9T^{2} \)
97 \( 1 + 5.60e3iT - 8.58e9T^{2} \)
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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

−16.34509240176150982922319270164, −15.80534720919889175043191920580, −14.86084045148042505458528333622, −13.27114341553847467508618757755, −12.24815938450507515158205909248, −10.08270425281129352650792385818, −8.379222036298079275947586539437, −7.21359043472801535211818053243, −6.19776650053271192005695497443, −4.03880605615288160342199233023, 0.32874473964203613859330521594, 3.02902707336217811850475304058, 4.28186441985192673122411633697, 7.11349721095983023606222529960, 9.326239388729532107278373000865, 10.17739082519017192331078100978, 11.58013183842730826387653650563, 12.48262627100472108766320247286, 13.52887679035699302975671804853, 15.61806719501370382185096850033

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