Properties

Label 2-2023-7.6-c0-0-1
Degree $2$
Conductor $2023$
Sign $1$
Analytic cond. $1.00960$
Root an. cond. $1.00479$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 7-s + 8-s + 9-s − 2·11-s + 14-s − 16-s − 18-s + 2·22-s + 23-s + 25-s + 29-s + 37-s − 43-s − 46-s + 49-s − 50-s − 53-s − 56-s − 58-s − 63-s + 64-s + 2·67-s + 71-s + 72-s − 74-s + 2·77-s + ⋯
L(s)  = 1  − 2-s − 7-s + 8-s + 9-s − 2·11-s + 14-s − 16-s − 18-s + 2·22-s + 23-s + 25-s + 29-s + 37-s − 43-s − 46-s + 49-s − 50-s − 53-s − 56-s − 58-s − 63-s + 64-s + 2·67-s + 71-s + 72-s − 74-s + 2·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2023} (1735, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5208637921\)
\(L(\frac12)\) \(\approx\) \(0.5208637921\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.482721435348213629570169264334, −8.554513802484796327171731539630, −7.897703535298147186120614035021, −7.18039252864529602902467907787, −6.49762536868650824522233983668, −5.17710524920520293304231624057, −4.63040178605986522361461475511, −3.33267995420614926597008141726, −2.34929078094694793167158853150, −0.823817223641608017608410817929, 0.823817223641608017608410817929, 2.34929078094694793167158853150, 3.33267995420614926597008141726, 4.63040178605986522361461475511, 5.17710524920520293304231624057, 6.49762536868650824522233983668, 7.18039252864529602902467907787, 7.897703535298147186120614035021, 8.554513802484796327171731539630, 9.482721435348213629570169264334

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