↑

Properties

Label 2-10-5.4-c9-0-1
Degree $2$
Conductor $10$
Sign $-0.998 - 0.0496i$
Analytic cond. $5.15035$
Root an. cond. $2.26944$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 16i·2-s + 254. i·3-s − 256·4-s + (69.4 − 1.39e3i)5-s − 4.07e3·6-s + 4.78e3i·7-s − 4.09e3i·8-s − 4.51e4·9-s + (2.23e4 + 1.11e3i)10-s + 2.26e4·11-s − 6.51e4i·12-s + 1.31e5i·13-s − 7.66e4·14-s + (3.55e5 + 1.76e4i)15-s + 6.55e4·16-s + 1.65e4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.81i·3-s − 0.5·4-s + (0.0496 − 0.998i)5-s − 1.28·6-s + 0.753i·7-s − 0.353i·8-s − 2.29·9-s + (0.706 + 0.0351i)10-s + 0.466·11-s − 0.907i·12-s + 1.27i·13-s − 0.532·14-s + (1.81 + 0.0901i)15-s + 0.250·16-s + 0.0479i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0496i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.998 - 0.0496i$
Analytic conductor: \(5.15035\)
Root analytic conductor: \(2.26944\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :9/2),\ -0.998 - 0.0496i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0313398 + 1.26101i\)
\(L(\frac12)\) \(\approx\) \(0.0313398 + 1.26101i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 16iT \)
5 \( 1 + (-69.4 + 1.39e3i)T \)
good3 \( 1 - 254. iT - 1.96e4T^{2} \)
7 \( 1 - 4.78e3iT - 4.03e7T^{2} \)
11 \( 1 - 2.26e4T + 2.35e9T^{2} \)
13 \( 1 - 1.31e5iT - 1.06e10T^{2} \)
17 \( 1 - 1.65e4iT - 1.18e11T^{2} \)
19 \( 1 - 1.73e5T + 3.22e11T^{2} \)
23 \( 1 - 2.08e6iT - 1.80e12T^{2} \)
29 \( 1 - 4.08e6T + 1.45e13T^{2} \)
31 \( 1 - 2.73e6T + 2.64e13T^{2} \)
37 \( 1 + 1.32e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.49e7T + 3.27e14T^{2} \)
43 \( 1 + 2.38e6iT - 5.02e14T^{2} \)
47 \( 1 + 2.41e7iT - 1.11e15T^{2} \)
53 \( 1 - 1.58e7iT - 3.29e15T^{2} \)
59 \( 1 - 9.17e7T + 8.66e15T^{2} \)
61 \( 1 + 9.24e7T + 1.16e16T^{2} \)
67 \( 1 - 5.16e7iT - 2.72e16T^{2} \)
71 \( 1 - 1.36e8T + 4.58e16T^{2} \)
73 \( 1 - 2.97e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.08e8T + 1.19e17T^{2} \)
83 \( 1 + 6.97e7iT - 1.86e17T^{2} \)
89 \( 1 - 7.43e8T + 3.50e17T^{2} \)
97 \( 1 + 1.60e9iT - 7.60e17T^{2} \)
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

−19.64928479919433012201785754214, −17.35244843317560568548686495762, −16.27683397419268823779758732962, −15.51883079116462957036136732492, −14.09851457127938498032226666775, −11.74836181911724604563497968015, −9.582877331672943819847063024252, −8.788909222408938161821611378684, −5.53510202738511668884936437692, −4.19085699587622456471185481557, 0.828568064698403981636625765331, 2.76927623526238644301959232882, 6.50605299324586431243291332122, 7.952005287620634517346020789149, 10.59987351316508526595084221549, 12.08183471228442366254244728310, 13.38092702072460507695123610426, 14.42287785238697031177321754114, 17.39894136145061632516088854823, 18.26037311529436129010063838912

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