Properties

Label 2-21-1.1-c3-0-2
Degree $2$
Conductor $21$
Sign $1$
Analytic cond. $1.23904$
Root an. cond. $1.11312$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 3·3-s + 8·4-s − 4·5-s − 12·6-s − 7·7-s + 9·9-s − 16·10-s + 62·11-s − 24·12-s − 62·13-s − 28·14-s + 12·15-s − 64·16-s + 84·17-s + 36·18-s + 100·19-s − 32·20-s + 21·21-s + 248·22-s − 42·23-s − 109·25-s − 248·26-s − 27·27-s − 56·28-s − 10·29-s + 48·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.357·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s − 0.505·10-s + 1.69·11-s − 0.577·12-s − 1.32·13-s − 0.534·14-s + 0.206·15-s − 16-s + 1.19·17-s + 0.471·18-s + 1.20·19-s − 0.357·20-s + 0.218·21-s + 2.40·22-s − 0.380·23-s − 0.871·25-s − 1.87·26-s − 0.192·27-s − 0.377·28-s − 0.0640·29-s + 0.292·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $1$
Analytic conductor: \(1.23904\)
Root analytic conductor: \(1.11312\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{21} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.652586624\)
\(L(\frac12)\) \(\approx\) \(1.652586624\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p T \)
7 \( 1 + p T \)
good2 \( 1 - p^{2} T + p^{3} T^{2} \)
5 \( 1 + 4 T + p^{3} T^{2} \)
11 \( 1 - 62 T + p^{3} T^{2} \)
13 \( 1 + 62 T + p^{3} T^{2} \)
17 \( 1 - 84 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 + 42 T + p^{3} T^{2} \)
29 \( 1 + 10 T + p^{3} T^{2} \)
31 \( 1 + 48 T + p^{3} T^{2} \)
37 \( 1 + 246 T + p^{3} T^{2} \)
41 \( 1 + 248 T + p^{3} T^{2} \)
43 \( 1 - 68 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 - 258 T + p^{3} T^{2} \)
59 \( 1 - 120 T + p^{3} T^{2} \)
61 \( 1 - 622 T + p^{3} T^{2} \)
67 \( 1 - 904 T + p^{3} T^{2} \)
71 \( 1 + 678 T + p^{3} T^{2} \)
73 \( 1 + 642 T + p^{3} T^{2} \)
79 \( 1 - 740 T + p^{3} T^{2} \)
83 \( 1 - 468 T + p^{3} T^{2} \)
89 \( 1 - 200 T + p^{3} T^{2} \)
97 \( 1 + 1266 T + p^{3} T^{2} \)
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

−17.40127996782650162168698349005, −16.20368624387748991924412491439, −14.85183094628299495439960961276, −13.88852072583434235810271714496, −12.18734969592006225633160749429, −11.84300799973327321430232327034, −9.655720314736560965383428982413, −7.01970943968790635781203522162, −5.46943372320388946773529265287, −3.78050195514546319844079133815, 3.78050195514546319844079133815, 5.46943372320388946773529265287, 7.01970943968790635781203522162, 9.655720314736560965383428982413, 11.84300799973327321430232327034, 12.18734969592006225633160749429, 13.88852072583434235810271714496, 14.85183094628299495439960961276, 16.20368624387748991924412491439, 17.40127996782650162168698349005

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