Properties

Label 2-29-1.1-c5-0-10
Degree $2$
Conductor $29$
Sign $-1$
Analytic cond. $4.65113$
Root an. cond. $2.15664$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.91·2-s − 18.2·3-s + 3.00·4-s − 97.7·5-s − 108.·6-s + 139.·7-s − 171.·8-s + 91.2·9-s − 578.·10-s + 533.·11-s − 54.9·12-s − 675.·13-s + 825.·14-s + 1.78e3·15-s − 1.11e3·16-s − 268.·17-s + 539.·18-s − 2.64e3·19-s − 293.·20-s − 2.55e3·21-s + 3.15e3·22-s + 794.·23-s + 3.13e3·24-s + 6.42e3·25-s − 3.99e3·26-s + 2.77e3·27-s + 419.·28-s + ⋯
L(s)  = 1  + 1.04·2-s − 1.17·3-s + 0.0939·4-s − 1.74·5-s − 1.22·6-s + 1.07·7-s − 0.947·8-s + 0.375·9-s − 1.82·10-s + 1.32·11-s − 0.110·12-s − 1.10·13-s + 1.12·14-s + 2.05·15-s − 1.08·16-s − 0.225·17-s + 0.392·18-s − 1.68·19-s − 0.164·20-s − 1.26·21-s + 1.38·22-s + 0.313·23-s + 1.11·24-s + 2.05·25-s − 1.16·26-s + 0.732·27-s + 0.101·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-1$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + 841T \)
good2 \( 1 - 5.91T + 32T^{2} \)
3 \( 1 + 18.2T + 243T^{2} \)
5 \( 1 + 97.7T + 3.12e3T^{2} \)
7 \( 1 - 139.T + 1.68e4T^{2} \)
11 \( 1 - 533.T + 1.61e5T^{2} \)
13 \( 1 + 675.T + 3.71e5T^{2} \)
17 \( 1 + 268.T + 1.41e6T^{2} \)
19 \( 1 + 2.64e3T + 2.47e6T^{2} \)
23 \( 1 - 794.T + 6.43e6T^{2} \)
31 \( 1 + 4.23e3T + 2.86e7T^{2} \)
37 \( 1 + 2.68e3T + 6.93e7T^{2} \)
41 \( 1 - 1.39e3T + 1.15e8T^{2} \)
43 \( 1 + 2.38e4T + 1.47e8T^{2} \)
47 \( 1 - 1.12e4T + 2.29e8T^{2} \)
53 \( 1 + 3.39e3T + 4.18e8T^{2} \)
59 \( 1 + 2.78e3T + 7.14e8T^{2} \)
61 \( 1 - 4.15e4T + 8.44e8T^{2} \)
67 \( 1 - 8.57e3T + 1.35e9T^{2} \)
71 \( 1 + 6.99e3T + 1.80e9T^{2} \)
73 \( 1 + 4.99e3T + 2.07e9T^{2} \)
79 \( 1 + 2.38e4T + 3.07e9T^{2} \)
83 \( 1 - 4.30e4T + 3.93e9T^{2} \)
89 \( 1 - 1.38e4T + 5.58e9T^{2} \)
97 \( 1 + 1.76e5T + 8.58e9T^{2} \)
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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

−15.06651560583581579760084873537, −14.58240123089896895347008893516, −12.51840770723615562574405252373, −11.80070269425034457392820771438, −11.12418386082560822681824002023, −8.549307801629179520181945231379, −6.79019200518989928730526496808, −4.97366964083269778478972425768, −4.06582300883637220147848087199, 0, 4.06582300883637220147848087199, 4.97366964083269778478972425768, 6.79019200518989928730526496808, 8.549307801629179520181945231379, 11.12418386082560822681824002023, 11.80070269425034457392820771438, 12.51840770723615562574405252373, 14.58240123089896895347008893516, 15.06651560583581579760084873537

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