↑

Properties

Label 2-30446-1.1-c1-0-5
Degree $2$
Conductor $30446$
Sign $-1$
Analytic cond. $243.112$
Root an. cond. $15.5920$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 5·7-s + 8-s − 2·9-s − 10-s + 5·11-s + 12-s − 13-s − 5·14-s − 15-s + 16-s + 2·17-s − 2·18-s + 2·19-s − 20-s − 5·21-s + 5·22-s + 4·23-s + 24-s − 4·25-s − 26-s − 5·27-s − 5·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.88·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.277·13-s − 1.33·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 0.458·19-s − 0.223·20-s − 1.09·21-s + 1.06·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.962·27-s − 0.944·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30446\)    =    \(2 \cdot 13 \cdot 1171\)
Sign: $-1$
Analytic conductor: \(243.112\)
Root analytic conductor: \(15.5920\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 30446,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
13 \( 1 + T \)
1171 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−15.12239069823443, −14.94194384113805, −14.26090658041566, −13.82931521423985, −13.28694726451877, −12.86526485217635, −12.16904399513430, −11.69466257740653, −11.55273667463844, −10.37139846897215, −10.07926631712545, −9.307570182581415, −9.008647794191255, −8.405425131759914, −7.480062894526599, −7.102293797196118, −6.432214322359850, −6.047337760654648, −5.387853966437742, −4.474690158969060, −3.806714624283687, −3.196966423248634, −3.129887090968123, −2.128391199888044, −1.064231693815547, 0, 1.064231693815547, 2.128391199888044, 3.129887090968123, 3.196966423248634, 3.806714624283687, 4.474690158969060, 5.387853966437742, 6.047337760654648, 6.432214322359850, 7.102293797196118, 7.480062894526599, 8.405425131759914, 9.008647794191255, 9.307570182581415, 10.07926631712545, 10.37139846897215, 11.55273667463844, 11.69466257740653, 12.16904399513430, 12.86526485217635, 13.28694726451877, 13.82931521423985, 14.26090658041566, 14.94194384113805, 15.12239069823443

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