↑

Properties

Label 2-10-5.2-c18-0-3
Degree $2$
Conductor $10$
Sign $0.800 - 0.599i$
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 + (1.07e4 − 1.07e4i)3-s + 1.31e5i·4-s + (−1.49e6 − 1.25e6i)5-s − 5.51e6·6-s + (3.13e7 + 3.13e7i)7-s + (3.35e7 − 3.35e7i)8-s + 1.55e8i·9-s + (6.34e7 + 7.04e8i)10-s − 4.34e9·11-s + (1.41e9 + 1.41e9i)12-s + (2.31e9 − 2.31e9i)13-s − 1.60e10i·14-s + (−2.96e10 + 2.66e9i)15-s − 1.71e10·16-s + (8.12e10 + 8.12e10i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.547 − 0.547i)3-s + 0.5i·4-s + (−0.767 − 0.640i)5-s − 0.547·6-s + (0.775 + 0.775i)7-s + (0.250 − 0.250i)8-s + 0.401i·9-s + (0.0634 + 0.704i)10-s − 1.84·11-s + (0.273 + 0.273i)12-s + (0.218 − 0.218i)13-s − 0.775i·14-s + (−0.770 + 0.0693i)15-s − 0.250·16-s + (0.684 + 0.684i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(20.5386\)
Root analytic conductor: \(4.53195\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :9),\ 0.800 - 0.599i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.06489 + 0.354882i\)
\(L(\frac12)\) \(\approx\) \(1.06489 + 0.354882i\)
\(L(10)\) 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 \( 1 + (1.49e6 + 1.25e6i)T \)
good3 \( 1 + (-1.07e4 + 1.07e4i)T - 3.87e8iT^{2} \)
7 \( 1 + (-3.13e7 - 3.13e7i)T + 1.62e15iT^{2} \)
11 \( 1 + 4.34e9T + 5.55e18T^{2} \)
13 \( 1 + (-2.31e9 + 2.31e9i)T - 1.12e20iT^{2} \)
17 \( 1 + (-8.12e10 - 8.12e10i)T + 1.40e22iT^{2} \)
19 \( 1 - 4.10e11iT - 1.04e23T^{2} \)
23 \( 1 + (-1.38e12 + 1.38e12i)T - 3.24e24iT^{2} \)
29 \( 1 - 3.56e12iT - 2.10e26T^{2} \)
31 \( 1 - 4.05e13T + 6.99e26T^{2} \)
37 \( 1 + (-8.12e13 - 8.12e13i)T + 1.68e28iT^{2} \)
41 \( 1 + 4.55e14T + 1.07e29T^{2} \)
43 \( 1 + (3.82e14 - 3.82e14i)T - 2.52e29iT^{2} \)
47 \( 1 + (9.27e13 + 9.27e13i)T + 1.25e30iT^{2} \)
53 \( 1 + (1.81e15 - 1.81e15i)T - 1.08e31iT^{2} \)
59 \( 1 - 9.16e15iT - 7.50e31T^{2} \)
61 \( 1 + 4.55e14T + 1.36e32T^{2} \)
67 \( 1 + (1.71e16 + 1.71e16i)T + 7.40e32iT^{2} \)
71 \( 1 - 1.60e16T + 2.10e33T^{2} \)
73 \( 1 + (6.23e15 - 6.23e15i)T - 3.46e33iT^{2} \)
79 \( 1 + 5.75e16iT - 1.43e34T^{2} \)
83 \( 1 + (7.69e16 - 7.69e16i)T - 3.49e34iT^{2} \)
89 \( 1 - 6.50e17iT - 1.22e35T^{2} \)
97 \( 1 + (-3.65e17 - 3.65e17i)T + 5.77e35iT^{2} \)
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

−16.55154002858756206603537691306, −15.12603241354992764245433941216, −13.19855392881726867527622901625, −12.12948753900292767434678829926, −10.52956118370265316777959995895, −8.226700870323448771528140592172, −8.053467869090684670377256073990, −5.03561630451554359078848888718, −2.82928006454225000478831845116, −1.39102136002067592279011173859, 0.47445733987204702563079817260, 2.99599693440209853529582181795, 4.77613893084194345990152272221, 7.18216294393549224835286139908, 8.262389641738340779194325807095, 10.05461790105297115725909447900, 11.27419522192258851060233245807, 13.75303187408271439122684447074, 15.10948132572028783548008166294, 15.84268350848791256616199882790

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