↑

Properties

Label 2-10-5.3-c8-0-3
Degree $2$
Conductor $10$
Sign $-0.598 + 0.801i$
Analytic cond. $4.07378$
Root an. cond. $2.01836$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (8 − 8i)2-s + (−39.7 − 39.7i)3-s − 128i·4-s + (−401. − 479. i)5-s − 636.·6-s + (144. − 144. i)7-s + (−1.02e3 − 1.02e3i)8-s − 3.39e3i·9-s + (−7.04e3 − 621. i)10-s + 1.36e4·11-s + (−5.09e3 + 5.09e3i)12-s + (3.08e4 + 3.08e4i)13-s − 2.31e3i·14-s + (−3.09e3 + 3.50e4i)15-s − 1.63e4·16-s + (4.94e3 − 4.94e3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.491 − 0.491i)3-s − 0.5i·4-s + (−0.642 − 0.766i)5-s − 0.491·6-s + (0.0601 − 0.0601i)7-s + (−0.250 − 0.250i)8-s − 0.517i·9-s + (−0.704 − 0.0621i)10-s + 0.928·11-s + (−0.245 + 0.245i)12-s + (1.08 + 1.08i)13-s − 0.0601i·14-s + (−0.0610 + 0.691i)15-s − 0.250·16-s + (0.0592 − 0.0592i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.598 + 0.801i$
Analytic conductor: \(4.07378\)
Root analytic conductor: \(2.01836\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :4),\ -0.598 + 0.801i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.637158 - 1.27134i\)
\(L(\frac12)\) \(\approx\) \(0.637158 - 1.27134i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8 + 8i)T \)
5 \( 1 + (401. + 479. i)T \)
good3 \( 1 + (39.7 + 39.7i)T + 6.56e3iT^{2} \)
7 \( 1 + (-144. + 144. i)T - 5.76e6iT^{2} \)
11 \( 1 - 1.36e4T + 2.14e8T^{2} \)
13 \( 1 + (-3.08e4 - 3.08e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-4.94e3 + 4.94e3i)T - 6.97e9iT^{2} \)
19 \( 1 + 1.76e5iT - 1.69e10T^{2} \)
23 \( 1 + (2.29e5 + 2.29e5i)T + 7.83e10iT^{2} \)
29 \( 1 + 1.44e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.50e6T + 8.52e11T^{2} \)
37 \( 1 + (1.60e6 - 1.60e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 3.62e6T + 7.98e12T^{2} \)
43 \( 1 + (5.75e5 + 5.75e5i)T + 1.16e13iT^{2} \)
47 \( 1 + (-3.11e6 + 3.11e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-8.16e6 - 8.16e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 1.54e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.92e7T + 1.91e14T^{2} \)
67 \( 1 + (1.00e7 - 1.00e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 1.84e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.45e7 - 2.45e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 2.17e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.00e6 - 4.00e6i)T + 2.25e15iT^{2} \)
89 \( 1 + 6.14e7iT - 3.93e15T^{2} \)
97 \( 1 + (-1.55e7 + 1.55e7i)T - 7.83e15iT^{2} \)
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

−18.74899178375951957395398157140, −17.15867894954235647710490256542, −15.62055112811890342928267547497, −13.78214722041872119100607104873, −12.25550210275362855451451000886, −11.42045660839704419276426396280, −8.986740064601155599554704885513, −6.48503423403572714390838654199, −4.21444491938491545766826703369, −0.986585465502458767950738782641, 3.82020611421012658503321505391, 5.96129765775866742918006259161, 7.946550086625866825545912688134, 10.53718074511801605687081798906, 11.92433843335304081474619900785, 13.92637399069533097122386065109, 15.34134078252824709278946876534, 16.35765836656535190721152744493, 17.88928497174011463984732795914, 19.51173591039567534973928063405

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