Properties

Label 2-103-1.1-c1-0-7
Degree $2$
Conductor $103$
Sign $-1$
Analytic cond. $0.822459$
Root an. cond. $0.906895$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.381·2-s − 3-s − 1.85·4-s − 2.61·5-s + 0.381·6-s − 7-s + 1.47·8-s − 2·9-s + 10-s − 0.381·11-s + 1.85·12-s + 1.85·13-s + 0.381·14-s + 2.61·15-s + 3.14·16-s − 3.38·17-s + 0.763·18-s − 0.854·19-s + 4.85·20-s + 21-s + 0.145·22-s − 4.47·23-s − 1.47·24-s + 1.85·25-s − 0.708·26-s + 5·27-s + 1.85·28-s + ⋯
L(s)  = 1  − 0.270·2-s − 0.577·3-s − 0.927·4-s − 1.17·5-s + 0.155·6-s − 0.377·7-s + 0.520·8-s − 0.666·9-s + 0.316·10-s − 0.115·11-s + 0.535·12-s + 0.514·13-s + 0.102·14-s + 0.675·15-s + 0.786·16-s − 0.820·17-s + 0.180·18-s − 0.195·19-s + 1.08·20-s + 0.218·21-s + 0.0311·22-s − 0.932·23-s − 0.300·24-s + 0.370·25-s − 0.138·26-s + 0.962·27-s + 0.350·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(103\)
Sign: $-1$
Analytic conductor: \(0.822459\)
Root analytic conductor: \(0.906895\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 103,\ (\ :1/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 + T \)
good2 \( 1 + 0.381T + 2T^{2} \)
3 \( 1 + T + 3T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 0.381T + 11T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 + 3.38T + 17T^{2} \)
19 \( 1 + 0.854T + 19T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 + 0.763T + 29T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 + 6.70T + 37T^{2} \)
41 \( 1 + 8.94T + 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 + 7.09T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 8.61T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 7.09T + 71T^{2} \)
73 \( 1 + 4.14T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 - 9.32T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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

−13.23610994267001705954558373175, −12.11379478170206293955745407270, −11.25361490978760122892096508652, −10.12933463243780987152001893331, −8.738425004717429255596542261962, −8.005816357257536141624131095437, −6.41211906014764542616529081381, −4.91231967485079938577327554470, −3.65104599359868815501951116461, 0, 3.65104599359868815501951116461, 4.91231967485079938577327554470, 6.41211906014764542616529081381, 8.005816357257536141624131095437, 8.738425004717429255596542261962, 10.12933463243780987152001893331, 11.25361490978760122892096508652, 12.11379478170206293955745407270, 13.23610994267001705954558373175

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