↑

Properties

Label 2-10-5.4-c11-0-3
Degree $2$
Conductor $10$
Sign $0.914 + 0.404i$
Analytic cond. $7.68343$
Root an. cond. $2.77190$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 32i·2-s − 100. i·3-s − 1.02e3·4-s + (2.82e3 − 6.39e3i)5-s + 3.23e3·6-s − 1.59e4i·7-s − 3.27e4i·8-s + 1.66e5·9-s + (2.04e5 + 9.04e4i)10-s + 6.23e5·11-s + 1.03e5i·12-s − 1.54e6i·13-s + 5.09e5·14-s + (−6.45e5 − 2.85e5i)15-s + 1.04e6·16-s − 1.11e6i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.239i·3-s − 0.5·4-s + (0.404 − 0.914i)5-s + 0.169·6-s − 0.357i·7-s − 0.353i·8-s + 0.942·9-s + (0.646 + 0.286i)10-s + 1.16·11-s + 0.119i·12-s − 1.15i·13-s + 0.252·14-s + (−0.219 − 0.0970i)15-s + 0.250·16-s − 0.191i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.914 + 0.404i$
Analytic conductor: \(7.68343\)
Root analytic conductor: \(2.77190\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :11/2),\ 0.914 + 0.404i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.69462 - 0.358105i\)
\(L(\frac12)\) \(\approx\) \(1.69462 - 0.358105i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 32iT \)
5 \( 1 + (-2.82e3 + 6.39e3i)T \)
good3 \( 1 + 100. iT - 1.77e5T^{2} \)
7 \( 1 + 1.59e4iT - 1.97e9T^{2} \)
11 \( 1 - 6.23e5T + 2.85e11T^{2} \)
13 \( 1 + 1.54e6iT - 1.79e12T^{2} \)
17 \( 1 + 1.11e6iT - 3.42e13T^{2} \)
19 \( 1 + 1.45e7T + 1.16e14T^{2} \)
23 \( 1 + 6.03e6iT - 9.52e14T^{2} \)
29 \( 1 - 2.91e7T + 1.22e16T^{2} \)
31 \( 1 - 2.33e8T + 2.54e16T^{2} \)
37 \( 1 - 6.65e8iT - 1.77e17T^{2} \)
41 \( 1 + 6.63e8T + 5.50e17T^{2} \)
43 \( 1 - 4.11e8iT - 9.29e17T^{2} \)
47 \( 1 - 2.47e9iT - 2.47e18T^{2} \)
53 \( 1 + 3.69e9iT - 9.26e18T^{2} \)
59 \( 1 + 1.25e9T + 3.01e19T^{2} \)
61 \( 1 + 4.05e9T + 4.35e19T^{2} \)
67 \( 1 - 1.84e10iT - 1.22e20T^{2} \)
71 \( 1 - 3.19e9T + 2.31e20T^{2} \)
73 \( 1 + 1.51e10iT - 3.13e20T^{2} \)
79 \( 1 - 4.26e10T + 7.47e20T^{2} \)
83 \( 1 - 5.86e10iT - 1.28e21T^{2} \)
89 \( 1 + 3.40e10T + 2.77e21T^{2} \)
97 \( 1 + 1.37e11iT - 7.15e21T^{2} \)
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

−17.68334895652407963655250177171, −16.74995913473419389300655639178, −15.29315655594229484003017574716, −13.60760447018549508473109581214, −12.47079417557992152820696623925, −9.951116983329592413735535243344, −8.310552652932346004264563907811, −6.46740506085338644090708872880, −4.49654193529319467939199623165, −1.05016682048521973650662562349, 1.92237251500110457091818055301, 4.04762458137670081151979365843, 6.61523985459222438030033949131, 9.202683005798838863213340954503, 10.55247796854966911380431734149, 12.04434708197507185530546201710, 13.81475835516542612802248691946, 15.11504954294462815155129364175, 17.06998159842014536558902476118, 18.56330199767593275809853417811

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