↑

Properties

Label 2-420-7.4-c1-0-1
Degree $2$
Conductor $420$
Sign $0.0633 - 0.997i$
Analytic cond. $3.35371$
Root an. cond. $1.83131$
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  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (1.32 + 2.29i)7-s + (−0.499 − 0.866i)9-s + (−1.82 + 3.15i)11-s + 2.64·13-s + 0.999·15-s + (−1.82 + 3.15i)17-s + (1.14 + 1.98i)19-s − 2.64·21-s + (1.82 + 3.15i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s + 2.35·29-s + (−3.14 + 5.44i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.499 + 0.866i)7-s + (−0.166 − 0.288i)9-s + (−0.549 + 0.951i)11-s + 0.733·13-s + 0.258·15-s + (−0.442 + 0.765i)17-s + (0.262 + 0.455i)19-s − 0.577·21-s + (0.380 + 0.658i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s + 0.437·29-s + (−0.564 + 0.978i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.827764 + 0.776899i\)
\(L(\frac12)\) \(\approx\) \(0.827764 + 0.776899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
good11 \( 1 + (1.82 - 3.15i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + (1.82 - 3.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.14 - 1.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.82 - 3.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + (3.14 - 5.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.32 - 4.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 - 9.93T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.64 + 6.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.46 + 4.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.32 + 4.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.35T + 71T^{2} \)
73 \( 1 + (-6.61 + 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.14 - 7.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.93T + 83T^{2} \)
89 \( 1 + (6.11 + 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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

−11.39324722022969155626835165251, −10.57042646794554647527285371058, −9.603033990183591555711841612396, −8.680817835565429436519200406498, −7.954484680292709139569487372073, −6.62238045031875506011021506259, −5.45321514715431897745615538167, −4.76656468176184778030563136089, −3.50245017863250232788486939605, −1.82413347989218123912622492691, 0.793861386719476647466411657677, 2.66057722699316317205439857229, 4.01394753110155522889990061745, 5.24448282593208848259950463580, 6.37060986667160101257639719278, 7.27890282793474291655420641581, 8.054038029956514139286885566531, 9.030569772573557562280142999767, 10.42377967640228773020390364148, 11.07030909276866190323689241054

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