Properties

Label 2-430-215.79-c1-0-20
Degree $2$
Conductor $430$
Sign $-0.878 + 0.476i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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  i·2-s + (2.75 − 1.59i)3-s − 4-s + (−2.23 − 0.0559i)5-s + (−1.59 − 2.75i)6-s + (−3.76 − 2.17i)7-s + i·8-s + (3.57 − 6.18i)9-s + (−0.0559 + 2.23i)10-s + 3.05·11-s + (−2.75 + 1.59i)12-s + (2.18 + 1.25i)13-s + (−2.17 + 3.76i)14-s + (−6.25 + 3.40i)15-s + 16-s + (−4.35 − 2.51i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (1.59 − 0.919i)3-s − 0.5·4-s + (−0.999 − 0.0250i)5-s + (−0.650 − 1.12i)6-s + (−1.42 − 0.822i)7-s + 0.353i·8-s + (1.19 − 2.06i)9-s + (−0.0176 + 0.706i)10-s + 0.921·11-s + (−0.796 + 0.459i)12-s + (0.604 + 0.349i)13-s + (−0.581 + 1.00i)14-s + (−1.61 + 0.879i)15-s + 0.250·16-s + (−1.05 − 0.609i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.878 + 0.476i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.878 + 0.476i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396020 - 1.56053i\)
\(L(\frac12)\) \(\approx\) \(0.396020 - 1.56053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (2.23 + 0.0559i)T \)
43 \( 1 + (4.82 + 4.44i)T \)
good3 \( 1 + (-2.75 + 1.59i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.76 + 2.17i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.05T + 11T^{2} \)
13 \( 1 + (-2.18 - 1.25i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.35 + 2.51i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.128 + 0.222i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.01 + 0.585i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.86 - 3.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.725 - 1.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.24 + 1.87i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 12.5T + 41T^{2} \)
47 \( 1 - 3.16iT - 47T^{2} \)
53 \( 1 + (-6.49 + 3.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.97T + 59T^{2} \)
61 \( 1 + (-5.56 + 9.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.17 + 4.72i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.80 - 13.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.16 - 1.24i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.92 + 6.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.5 - 7.22i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.69 - 6.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.7iT - 97T^{2} \)
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

−10.85648169822536727451729554171, −9.507010651967265555947949244932, −9.078560750157439677346961693270, −8.181675484633732169992533409391, −7.04773272838792336084246575922, −6.66958504493311580890551433616, −4.07053536225436880268299496847, −3.64279078182402047161657273102, −2.59114373664757504149999194877, −0.908627070334034118798993297365, 2.74093294441097336338693372669, 3.72764015622164936700320284674, 4.32194049107914411678866972150, 5.99321729766350533787794629587, 7.06242639154579256931038602359, 8.163070583535539029534219632463, 8.832605378629165845334039730214, 9.329531993718462871071111440558, 10.24301796406322258296728900241, 11.46455226249157776340439874333

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