Properties

Label 2-1-1.1-c109-0-2
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $75.2394$
Root an. cond. $8.67406$
Motivic weight $109$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.02e16·2-s − 7.88e25·3-s − 2.40e32·4-s + 1.60e38·5-s + 1.59e42·6-s + 1.73e45·7-s + 1.79e49·8-s − 3.92e51·9-s − 3.24e54·10-s − 1.26e56·11-s + 1.89e58·12-s − 6.03e60·13-s − 3.50e61·14-s − 1.26e64·15-s − 2.07e65·16-s − 1.23e67·17-s + 7.93e67·18-s + 8.35e69·19-s − 3.86e70·20-s − 1.36e71·21-s + 2.55e72·22-s + 1.30e74·23-s − 1.41e75·24-s + 1.04e76·25-s + 1.21e77·26-s + 1.10e78·27-s − 4.17e77·28-s + ⋯
L(s)  = 1  − 0.793·2-s − 0.782·3-s − 0.370·4-s + 1.29·5-s + 0.620·6-s + 0.151·7-s + 1.08·8-s − 0.387·9-s − 1.02·10-s − 0.221·11-s + 0.290·12-s − 1.17·13-s − 0.120·14-s − 1.01·15-s − 0.491·16-s − 1.07·17-s + 0.307·18-s + 1.69·19-s − 0.480·20-s − 0.118·21-s + 0.175·22-s + 0.797·23-s − 0.851·24-s + 0.676·25-s + 0.933·26-s + 1.08·27-s − 0.0563·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(110-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+54.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(75.2394\)
Root analytic conductor: \(8.67406\)
Motivic weight: \(109\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :109/2),\ -1)\)

Particular Values

\(L(55)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{111}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.02e16T + 6.49e32T^{2} \)
3 \( 1 + 7.88e25T + 1.01e52T^{2} \)
5 \( 1 - 1.60e38T + 1.54e76T^{2} \)
7 \( 1 - 1.73e45T + 1.30e92T^{2} \)
11 \( 1 + 1.26e56T + 3.24e113T^{2} \)
13 \( 1 + 6.03e60T + 2.62e121T^{2} \)
17 \( 1 + 1.23e67T + 1.31e134T^{2} \)
19 \( 1 - 8.35e69T + 2.42e139T^{2} \)
23 \( 1 - 1.30e74T + 2.68e148T^{2} \)
29 \( 1 + 1.12e79T + 2.51e159T^{2} \)
31 \( 1 + 1.73e81T + 3.61e162T^{2} \)
37 \( 1 - 3.88e85T + 8.58e170T^{2} \)
41 \( 1 - 7.86e87T + 6.21e175T^{2} \)
43 \( 1 - 3.55e88T + 1.11e178T^{2} \)
47 \( 1 + 2.02e90T + 1.81e182T^{2} \)
53 \( 1 - 1.36e94T + 8.83e187T^{2} \)
59 \( 1 - 1.85e96T + 1.05e193T^{2} \)
61 \( 1 + 1.53e97T + 3.98e194T^{2} \)
67 \( 1 + 1.76e99T + 1.10e199T^{2} \)
71 \( 1 - 1.19e101T + 6.12e201T^{2} \)
73 \( 1 + 5.70e101T + 1.26e203T^{2} \)
79 \( 1 + 3.23e103T + 6.93e206T^{2} \)
83 \( 1 + 7.45e104T + 1.51e209T^{2} \)
89 \( 1 - 2.08e106T + 3.04e212T^{2} \)
97 \( 1 + 1.80e107T + 3.61e216T^{2} \)
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

−13.06752390717417469834278451782, −11.24359784541137692329556531130, −9.943655614927464452992610416443, −9.058679387804406145567211830831, −7.32208622996020121582320349393, −5.70089867431840766010910606129, −4.81944974173765843683828084456, −2.49755491621002396002902594511, −1.13876916935493967162061779737, 0, 1.13876916935493967162061779737, 2.49755491621002396002902594511, 4.81944974173765843683828084456, 5.70089867431840766010910606129, 7.32208622996020121582320349393, 9.058679387804406145567211830831, 9.943655614927464452992610416443, 11.24359784541137692329556531130, 13.06752390717417469834278451782

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