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Properties

Label 1-2153-2153.2152-r0-0-0
Degree $1$
Conductor $2153$
Sign $1$
Analytic cond. $9.99848$
Root an. cond. $9.99848$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s + 18-s + 19-s − 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s + 18-s + 19-s − 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2153\)
Sign: $1$
Analytic conductor: \(9.99848\)
Root analytic conductor: \(9.99848\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2153} (2152, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2153,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.920326933\)
\(L(\frac12)\) \(\approx\) \(1.920326933\)
\(L(1)\) \(\approx\) \(1.323239689\)
\(L(1)\) \(\approx\) \(1.323239689\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2153 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.95145543211223687061061984103, −19.20547517976456695624006003895, −18.11761299905171788402777522799, −17.69831867458352452577464821543, −16.66054490976242951759332844185, −15.97724057690381084708641921408, −15.50611409896634157853187013016, −14.825532877121668192191940910390, −13.94527322857627929804819044986, −13.10007928947044359423773617498, −12.280213669725368349595803259651, −11.79843451459060790331708466238, −11.23953515621428633042496750553, −10.59012760580968094452643331263, −9.78156594504130736685264467891, −8.1042632072642131491042122790, −7.75122263113412922281173265087, −6.94063342931697627395314283564, −6.1147791464873360489162510053, −4.9599603236773730375472918047, −4.85802608263784885897961010069, −4.071946644008399756261619913978, −2.88991912054729478707410640907, −1.99607267348483131817005523565, −0.76044685145887917438475242026, 0.76044685145887917438475242026, 1.99607267348483131817005523565, 2.88991912054729478707410640907, 4.071946644008399756261619913978, 4.85802608263784885897961010069, 4.9599603236773730375472918047, 6.1147791464873360489162510053, 6.94063342931697627395314283564, 7.75122263113412922281173265087, 8.1042632072642131491042122790, 9.78156594504130736685264467891, 10.59012760580968094452643331263, 11.23953515621428633042496750553, 11.79843451459060790331708466238, 12.280213669725368349595803259651, 13.10007928947044359423773617498, 13.94527322857627929804819044986, 14.825532877121668192191940910390, 15.50611409896634157853187013016, 15.97724057690381084708641921408, 16.66054490976242951759332844185, 17.69831867458352452577464821543, 18.11761299905171788402777522799, 19.20547517976456695624006003895, 19.95145543211223687061061984103

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