Properties

Label 2-1-1.1-c35-0-1
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $7.75951$
Root an. cond. $2.78559$
Motivic weight $35$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.68e4·2-s + 3.95e8·3-s − 3.36e10·4-s + 8.21e11·5-s − 1.06e13·6-s + 6.06e14·7-s + 1.82e15·8-s + 1.06e17·9-s − 2.20e16·10-s + 1.23e18·11-s − 1.33e19·12-s − 8.46e17·13-s − 1.62e19·14-s + 3.25e20·15-s + 1.10e21·16-s − 3.92e21·17-s − 2.85e21·18-s − 3.52e20·19-s − 2.76e22·20-s + 2.40e23·21-s − 3.30e22·22-s − 8.58e23·23-s + 7.21e23·24-s − 2.23e24·25-s + 2.26e22·26-s + 2.23e25·27-s − 2.04e25·28-s + ⋯
L(s)  = 1  − 0.144·2-s + 1.76·3-s − 0.979·4-s + 0.481·5-s − 0.255·6-s + 0.986·7-s + 0.286·8-s + 2.13·9-s − 0.0696·10-s + 0.735·11-s − 1.73·12-s − 0.0271·13-s − 0.142·14-s + 0.851·15-s + 0.937·16-s − 1.14·17-s − 0.308·18-s − 0.0147·19-s − 0.471·20-s + 1.74·21-s − 0.106·22-s − 1.26·23-s + 0.506·24-s − 0.768·25-s + 0.00392·26-s + 1.99·27-s − 0.965·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(36-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+35/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(7.75951\)
Root analytic conductor: \(2.78559\)
Motivic weight: \(35\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :35/2),\ 1)\)

Particular Values

\(L(18)\) \(\approx\) \(2.558058893\)
\(L(\frac12)\) \(\approx\) \(2.558058893\)
\(L(\frac{37}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.68e4T + 3.43e10T^{2} \)
3 \( 1 - 3.95e8T + 5.00e16T^{2} \)
5 \( 1 - 8.21e11T + 2.91e24T^{2} \)
7 \( 1 - 6.06e14T + 3.78e29T^{2} \)
11 \( 1 - 1.23e18T + 2.81e36T^{2} \)
13 \( 1 + 8.46e17T + 9.72e38T^{2} \)
17 \( 1 + 3.92e21T + 1.16e43T^{2} \)
19 \( 1 + 3.52e20T + 5.70e44T^{2} \)
23 \( 1 + 8.58e23T + 4.57e47T^{2} \)
29 \( 1 + 1.38e25T + 1.52e51T^{2} \)
31 \( 1 + 3.33e25T + 1.57e52T^{2} \)
37 \( 1 - 1.99e27T + 7.71e54T^{2} \)
41 \( 1 - 2.10e28T + 2.80e56T^{2} \)
43 \( 1 + 4.57e28T + 1.48e57T^{2} \)
47 \( 1 + 1.72e29T + 3.33e58T^{2} \)
53 \( 1 - 1.81e30T + 2.23e60T^{2} \)
59 \( 1 + 5.02e30T + 9.54e61T^{2} \)
61 \( 1 + 3.25e29T + 3.06e62T^{2} \)
67 \( 1 + 4.01e31T + 8.17e63T^{2} \)
71 \( 1 - 3.85e32T + 6.22e64T^{2} \)
73 \( 1 - 9.93e31T + 1.64e65T^{2} \)
79 \( 1 + 2.62e33T + 2.61e66T^{2} \)
83 \( 1 - 6.17e33T + 1.47e67T^{2} \)
89 \( 1 - 6.75e33T + 1.69e68T^{2} \)
97 \( 1 + 8.10e34T + 3.44e69T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.51213991249091562445044194612, −21.60990521007096499237175467177, −19.84115990496816626692568244831, −18.00626150863511425985242461230, −14.63850500192579912510421319527, −13.54187465949206101269929955780, −9.459703826379898497888748758919, −8.157521796120929573693857359706, −4.13950803345000955418253124047, −1.83589775632061795320516641947, 1.83589775632061795320516641947, 4.13950803345000955418253124047, 8.157521796120929573693857359706, 9.459703826379898497888748758919, 13.54187465949206101269929955780, 14.63850500192579912510421319527, 18.00626150863511425985242461230, 19.84115990496816626692568244831, 21.60990521007096499237175467177, 24.51213991249091562445044194612

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