↑

Properties

Label 2-430-43.36-c1-0-15
Degree $2$
Conductor $430$
Sign $-0.0861 - 0.996i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1.5 − 2.59i)3-s + 4-s + (−0.5 − 0.866i)5-s + (1.5 + 2.59i)6-s − 8-s + (−3 + 5.19i)9-s + (0.5 + 0.866i)10-s − 6·11-s + (−1.5 − 2.59i)12-s + (−1.5 + 2.59i)15-s + 16-s + (2 − 3.46i)17-s + (3 − 5.19i)18-s + (2 + 3.46i)19-s + (−0.5 − 0.866i)20-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.866 − 1.49i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (0.612 + 1.06i)6-s − 0.353·8-s + (−1 + 1.73i)9-s + (0.158 + 0.273i)10-s − 1.80·11-s + (−0.433 − 0.749i)12-s + (−0.387 + 0.670i)15-s + 0.250·16-s + (0.485 − 0.840i)17-s + (0.707 − 1.22i)18-s + (0.458 + 0.794i)19-s + (−0.111 − 0.193i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0861 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0861 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.0861 - 0.996i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.0861 - 0.996i)\)

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 \( 1 + T \)
5 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (6.5 + 0.866i)T \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4 - 6.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4 - 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.5 + 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \)
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

−10.68806091808231158796079668089, −9.647153288400384047556688445816, −8.249860879405008774862463864305, −7.70392866215363538328473552958, −7.03941804960068795595043853125, −5.78867838504659961623646506543, −5.14456401471591198803257239420, −2.83020870303270225348206797819, −1.44056956029966402223138470886, 0, 2.77937321113669562370556187255, 4.03152599555149067529829374220, 5.25767330633036476134860173760, 5.94134191805903154512109648994, 7.34390571778061140995842898880, 8.292209352422757863040016787814, 9.472298557554296097755407749912, 10.11740796518777914487880407003, 10.82329198021321592103326677486

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