↑

Properties

Label 2-430-43.6-c1-0-7
Degree $2$
Conductor $430$
Sign $0.955 - 0.295i$
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  + 2-s + (0.339 − 0.587i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.339 − 0.587i)6-s + (1 + 1.73i)7-s + 8-s + (1.26 + 2.19i)9-s + (−0.5 + 0.866i)10-s − 0.321·11-s + (0.339 − 0.587i)12-s + (2.10 + 3.65i)13-s + (1 + 1.73i)14-s + (0.339 + 0.587i)15-s + 16-s + (−2.94 − 5.10i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.195 − 0.339i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.138 − 0.239i)6-s + (0.377 + 0.654i)7-s + 0.353·8-s + (0.423 + 0.733i)9-s + (−0.158 + 0.273i)10-s − 0.0969·11-s + (0.0979 − 0.169i)12-s + (0.584 + 1.01i)13-s + (0.267 + 0.462i)14-s + (0.0875 + 0.151i)15-s + 0.250·16-s + (−0.715 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.955 - 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26360 + 0.342434i\)
\(L(\frac12)\) \(\approx\) \(2.26360 + 0.342434i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (3.25 + 5.69i)T \)
good3 \( 1 + (-0.339 + 0.587i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.321T + 11T^{2} \)
13 \( 1 + (-2.10 - 3.65i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.94 + 5.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.05 + 5.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 + 2.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.55 + 4.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.10 - 5.38i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.78 - 6.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.71T + 41T^{2} \)
47 \( 1 + 2.89T + 47T^{2} \)
53 \( 1 + (2.89 - 5.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.18T + 59T^{2} \)
61 \( 1 + (4.86 + 8.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.82 - 8.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.43 + 2.47i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.51 - 6.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.643 - 1.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.87 + 10.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.39 + 9.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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

−11.45509548256058640262512532822, −10.60652852135284528322219150704, −9.294656111988842332498594677665, −8.399307316512088631645735919917, −7.19904832129510990098408758545, −6.70929693707177321158612682195, −5.25023562661139028296335697177, −4.50404813096145244081434445667, −2.99269047899058335543651051300, −1.93942175396529405084998408994, 1.44326886975905127056968349797, 3.49426812132176771485635050522, 4.00742520599649239812187625235, 5.25522798582439661358770473369, 6.23124894619503334488260192772, 7.45086684702236960942653295527, 8.243931427961865669570121172183, 9.373544927390759800917290024109, 10.43946978003522053325569607920, 11.04327002695375944132538320035

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