Properties

Label 2-103-1.1-c1-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s − 3-s + 4.85·4-s − 0.381·5-s + 2.61·6-s − 7-s − 7.47·8-s − 2·9-s + 10-s − 2.61·11-s − 4.85·12-s − 4.85·13-s + 2.61·14-s + 0.381·15-s + 9.85·16-s − 5.61·17-s + 5.23·18-s + 5.85·19-s − 1.85·20-s + 21-s + 6.85·22-s + 4.47·23-s + 7.47·24-s − 4.85·25-s + 12.7·26-s + 5·27-s − 4.85·28-s + ⋯
L(s)  = 1  − 1.85·2-s − 0.577·3-s + 2.42·4-s − 0.170·5-s + 1.06·6-s − 0.377·7-s − 2.64·8-s − 0.666·9-s + 0.316·10-s − 0.789·11-s − 1.40·12-s − 1.34·13-s + 0.699·14-s + 0.0986·15-s + 2.46·16-s − 1.36·17-s + 1.23·18-s + 1.34·19-s − 0.414·20-s + 0.218·21-s + 1.46·22-s + 0.932·23-s + 1.52·24-s − 0.970·25-s + 2.49·26-s + 0.962·27-s − 0.917·28-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 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 + T + 3T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 2.61T + 11T^{2} \)
13 \( 1 + 4.85T + 13T^{2} \)
17 \( 1 + 5.61T + 17T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 5.23T + 29T^{2} \)
31 \( 1 + 6.70T + 31T^{2} \)
37 \( 1 - 6.70T + 37T^{2} \)
41 \( 1 - 8.94T + 41T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 - 4.09T + 47T^{2} \)
53 \( 1 - 1.09T + 53T^{2} \)
59 \( 1 - 6.38T + 59T^{2} \)
61 \( 1 - 4.14T + 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 + 4.09T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 6.56T + 79T^{2} \)
83 \( 1 + 6.32T + 83T^{2} \)
89 \( 1 + 2.29T + 89T^{2} \)
97 \( 1 + 1.70T + 97T^{2} \)
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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

−12.94840009397894493978095356198, −11.60864535731955206900512239723, −11.08722419550826236265644041303, −9.888856913147113375102993549612, −9.102211533828713010939301018281, −7.80272369775436991917087217809, −6.94244585770451458914106171343, −5.49393216015653352470861077787, −2.59208883639954584899311384770, 0, 2.59208883639954584899311384770, 5.49393216015653352470861077787, 6.94244585770451458914106171343, 7.80272369775436991917087217809, 9.102211533828713010939301018281, 9.888856913147113375102993549612, 11.08722419550826236265644041303, 11.60864535731955206900512239723, 12.94840009397894493978095356198

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