Properties

Degree 2
Conductor 103
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·2-s − 0.618·4-s − 1.61·7-s − 8-s + 9-s + 0.618·13-s − 1.00·14-s − 1.61·17-s + 0.618·18-s + 0.618·19-s + 0.618·23-s + 25-s + 0.381·26-s + 0.999·28-s − 1.61·29-s + 0.999·32-s − 1.00·34-s − 0.618·36-s + 0.381·38-s + 0.618·41-s + 0.381·46-s + 1.61·49-s + 0.618·50-s − 0.381·52-s + 1.61·56-s − 1.00·58-s − 1.61·59-s + ⋯
L(s)  = 1  + 0.618·2-s − 0.618·4-s − 1.61·7-s − 8-s + 9-s + 0.618·13-s − 1.00·14-s − 1.61·17-s + 0.618·18-s + 0.618·19-s + 0.618·23-s + 25-s + 0.381·26-s + 0.999·28-s − 1.61·29-s + 0.999·32-s − 1.00·34-s − 0.618·36-s + 0.381·38-s + 0.618·41-s + 0.381·46-s + 1.61·49-s + 0.618·50-s − 0.381·52-s + 1.61·56-s − 1.00·58-s − 1.61·59-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{103} (102, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 103,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.6119819366\)
\(L(\frac12)\)  \(\approx\)  \(0.6119819366\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 103$,\(F_p(T)\) is a polynomial of degree 2. If $p = 103$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad103 \( 1 - T \)
good2 \( 1 - 0.618T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 + 1.61T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 + 1.61T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 0.618T + T^{2} \)
83 \( 1 + 1.61T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.68380576693065448866038657735, −13.06999783634067273401280863882, −12.51791807552834908624905200911, −10.91161638857600902781983446815, −9.587760247432759093520355809873, −8.974746013144302001889982996005, −7.07266036729859331897519459089, −6.03454633619783779298064033402, −4.46895357445124374556423870030, −3.28396214040166531134869327212, 3.28396214040166531134869327212, 4.46895357445124374556423870030, 6.03454633619783779298064033402, 7.07266036729859331897519459089, 8.974746013144302001889982996005, 9.587760247432759093520355809873, 10.91161638857600902781983446815, 12.51791807552834908624905200911, 13.06999783634067273401280863882, 13.68380576693065448866038657735

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