↑

Properties

Label 2-10-5.3-c18-0-0
Degree $2$
Conductor $10$
Sign $0.338 - 0.940i$
Analytic cond. $20.5386$
Root an. cond. $4.53195$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (256 − 256i)2-s + (−1.31e4 − 1.31e4i)3-s − 1.31e5i·4-s + (−1.63e6 − 1.06e6i)5-s − 6.72e6·6-s + (−1.89e7 + 1.89e7i)7-s + (−3.35e7 − 3.35e7i)8-s − 4.21e7i·9-s + (−6.91e8 + 1.46e8i)10-s − 2.85e8·11-s + (−1.72e9 + 1.72e9i)12-s + (5.93e9 + 5.93e9i)13-s + 9.72e9i·14-s + (7.49e9 + 3.55e10i)15-s − 1.71e10·16-s + (1.10e10 − 1.10e10i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.667 − 0.667i)3-s − 0.5i·4-s + (−0.837 − 0.545i)5-s − 0.667·6-s + (−0.470 + 0.470i)7-s + (−0.250 − 0.250i)8-s − 0.108i·9-s + (−0.691 + 0.146i)10-s − 0.121·11-s + (−0.333 + 0.333i)12-s + (0.559 + 0.559i)13-s + 0.470i·14-s + (0.194 + 0.923i)15-s − 0.250·16-s + (0.0934 − 0.0934i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.338 - 0.940i$
Analytic conductor: \(20.5386\)
Root analytic conductor: \(4.53195\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :9),\ 0.338 - 0.940i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.158007 + 0.111047i\)
\(L(\frac12)\) \(\approx\) \(0.158007 + 0.111047i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-256 + 256i)T \)
5 \( 1 + (1.63e6 + 1.06e6i)T \)
good3 \( 1 + (1.31e4 + 1.31e4i)T + 3.87e8iT^{2} \)
7 \( 1 + (1.89e7 - 1.89e7i)T - 1.62e15iT^{2} \)
11 \( 1 + 2.85e8T + 5.55e18T^{2} \)
13 \( 1 + (-5.93e9 - 5.93e9i)T + 1.12e20iT^{2} \)
17 \( 1 + (-1.10e10 + 1.10e10i)T - 1.40e22iT^{2} \)
19 \( 1 + 8.45e10iT - 1.04e23T^{2} \)
23 \( 1 + (-1.46e12 - 1.46e12i)T + 3.24e24iT^{2} \)
29 \( 1 - 1.81e13iT - 2.10e26T^{2} \)
31 \( 1 + 2.03e13T + 6.99e26T^{2} \)
37 \( 1 + (-1.14e14 + 1.14e14i)T - 1.68e28iT^{2} \)
41 \( 1 + 5.54e14T + 1.07e29T^{2} \)
43 \( 1 + (6.60e14 + 6.60e14i)T + 2.52e29iT^{2} \)
47 \( 1 + (-1.02e14 + 1.02e14i)T - 1.25e30iT^{2} \)
53 \( 1 + (3.27e15 + 3.27e15i)T + 1.08e31iT^{2} \)
59 \( 1 - 1.17e16iT - 7.50e31T^{2} \)
61 \( 1 - 3.17e15T + 1.36e32T^{2} \)
67 \( 1 + (2.09e16 - 2.09e16i)T - 7.40e32iT^{2} \)
71 \( 1 - 1.64e16T + 2.10e33T^{2} \)
73 \( 1 + (5.59e16 + 5.59e16i)T + 3.46e33iT^{2} \)
79 \( 1 - 1.72e17iT - 1.43e34T^{2} \)
83 \( 1 + (-1.68e17 - 1.68e17i)T + 3.49e34iT^{2} \)
89 \( 1 - 5.70e17iT - 1.22e35T^{2} \)
97 \( 1 + (-5.32e17 + 5.32e17i)T - 5.77e35iT^{2} \)
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

−16.53465244050730550383193334805, −15.18189803750294493650918109766, −13.18889701820433825165978304150, −12.20495598064202034124337218826, −11.24060450985369727357030756557, −9.039135061421223951530773322631, −6.92880576789217606453758376314, −5.34325244721607867570598425011, −3.52114971534989512056362420651, −1.33903331625332319682600551766, 0.06858847862241754604450394187, 3.32354989097256476282915057206, 4.67526485487386385552774395145, 6.36349963453425519975244093573, 7.952321303780381163580413010371, 10.31818622943140442525917089024, 11.50509420717209216629049697742, 13.21681765411229488593157838693, 14.94309158194469899781176404258, 15.99671973025082796673712955490

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