↑

Properties

Label 2-429-143.21-c1-0-19
Degree $2$
Conductor $429$
Sign $0.949 + 0.314i$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.10 − 1.10i)2-s + 3-s − 0.441i·4-s + (1.95 + 1.95i)5-s + (1.10 − 1.10i)6-s + (−1.11 − 1.11i)7-s + (1.72 + 1.72i)8-s + 9-s + 4.32·10-s + (−1.55 + 2.92i)11-s − 0.441i·12-s + (−1.60 − 3.22i)13-s − 2.46·14-s + (1.95 + 1.95i)15-s + 4.68·16-s + 0.0613·17-s + ⋯
L(s)  = 1  + (0.781 − 0.781i)2-s + 0.577·3-s − 0.220i·4-s + (0.874 + 0.874i)5-s + (0.451 − 0.451i)6-s + (−0.421 − 0.421i)7-s + (0.608 + 0.608i)8-s + 0.333·9-s + 1.36·10-s + (−0.469 + 0.883i)11-s − 0.127i·12-s + (−0.446 − 0.894i)13-s − 0.658·14-s + (0.504 + 0.504i)15-s + 1.17·16-s + 0.0148·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.949 + 0.314i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.949 + 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.64154 - 0.426181i\)
\(L(\frac12)\) \(\approx\) \(2.64154 - 0.426181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 + (1.55 - 2.92i)T \)
13 \( 1 + (1.60 + 3.22i)T \)
good2 \( 1 + (-1.10 + 1.10i)T - 2iT^{2} \)
5 \( 1 + (-1.95 - 1.95i)T + 5iT^{2} \)
7 \( 1 + (1.11 + 1.11i)T + 7iT^{2} \)
17 \( 1 - 0.0613T + 17T^{2} \)
19 \( 1 + (1.53 - 1.53i)T - 19iT^{2} \)
23 \( 1 + 6.25iT - 23T^{2} \)
29 \( 1 + 5.40iT - 29T^{2} \)
31 \( 1 + (1.00 + 1.00i)T + 31iT^{2} \)
37 \( 1 + (1.41 - 1.41i)T - 37iT^{2} \)
41 \( 1 + (-6.30 + 6.30i)T - 41iT^{2} \)
43 \( 1 + 6.96T + 43T^{2} \)
47 \( 1 + (6.90 - 6.90i)T - 47iT^{2} \)
53 \( 1 + 6.29T + 53T^{2} \)
59 \( 1 + (-8.28 + 8.28i)T - 59iT^{2} \)
61 \( 1 - 9.57iT - 61T^{2} \)
67 \( 1 + (-2.56 - 2.56i)T + 67iT^{2} \)
71 \( 1 + (-2.26 - 2.26i)T + 71iT^{2} \)
73 \( 1 + (3.20 + 3.20i)T + 73iT^{2} \)
79 \( 1 - 7.07iT - 79T^{2} \)
83 \( 1 + (9.65 - 9.65i)T - 83iT^{2} \)
89 \( 1 + (-4.26 + 4.26i)T - 89iT^{2} \)
97 \( 1 + (-9.43 - 9.43i)T + 97iT^{2} \)
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

−11.00474942504765150729486307714, −10.18249790324611823431777835324, −9.882606278524348467923353196324, −8.294353082906497266647208781898, −7.40204178989047380447273415352, −6.36640581087973628788281614460, −5.07743467019455974029211941494, −3.95102535388647432243083464759, −2.80506622735271060768190616251, −2.17485410349867438151198435387, 1.67598784154581839500726330287, 3.30574805626386982695473341155, 4.70252360400393390855070902743, 5.46186088626434986223438415730, 6.30445357133105172522207530470, 7.31279972888821232256810779801, 8.522777195965283144078075142345, 9.329967381226436559339766424737, 9.987728640938404861146483350233, 11.28515841322003066894786651241

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