Properties

Label 2-2093-1.1-c1-0-74
Degree $2$
Conductor $2093$
Sign $1$
Analytic cond. $16.7126$
Root an. cond. $4.08811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 2·4-s + 3·5-s − 6·6-s − 7-s + 6·9-s − 6·10-s + 3·11-s + 6·12-s − 13-s + 2·14-s + 9·15-s − 4·16-s + 4·17-s − 12·18-s + 5·19-s + 6·20-s − 3·21-s − 6·22-s + 23-s + 4·25-s + 2·26-s + 9·27-s − 2·28-s − 2·29-s − 18·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 4-s + 1.34·5-s − 2.44·6-s − 0.377·7-s + 2·9-s − 1.89·10-s + 0.904·11-s + 1.73·12-s − 0.277·13-s + 0.534·14-s + 2.32·15-s − 16-s + 0.970·17-s − 2.82·18-s + 1.14·19-s + 1.34·20-s − 0.654·21-s − 1.27·22-s + 0.208·23-s + 4/5·25-s + 0.392·26-s + 1.73·27-s − 0.377·28-s − 0.371·29-s − 3.28·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2093 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2093 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2093\)    =    \(7 \cdot 13 \cdot 23\)
Sign: $1$
Analytic conductor: \(16.7126\)
Root analytic conductor: \(4.08811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2093} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2093,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.231906784\)
\(L(\frac12)\) \(\approx\) \(2.231906784\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
13 \( 1 + T \)
23 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 - p T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−9.222019561151958328426484517241, −8.688826821180598629493645375650, −7.64655320613516909719858259583, −7.28844766733467800592655040050, −6.30993182380270345792594627084, −5.17454361285411116769376627245, −3.78773020802889488322340342274, −2.92681568518083353935422532485, −1.91848986629275014460102648896, −1.30559497828824558702268779820, 1.30559497828824558702268779820, 1.91848986629275014460102648896, 2.92681568518083353935422532485, 3.78773020802889488322340342274, 5.17454361285411116769376627245, 6.30993182380270345792594627084, 7.28844766733467800592655040050, 7.64655320613516909719858259583, 8.688826821180598629493645375650, 9.222019561151958328426484517241

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