↑

Properties

Label 2-10-5.3-c16-0-6
Degree $2$
Conductor $10$
Sign $-0.935 + 0.353i$
Analytic cond. $16.2324$
Root an. cond. $4.02895$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (128 − 128i)2-s + (2.24e3 + 2.24e3i)3-s − 3.27e4i·4-s + (−5.05e4 − 3.87e5i)5-s + 5.75e5·6-s + (−5.28e6 + 5.28e6i)7-s + (−4.19e6 − 4.19e6i)8-s − 3.29e7i·9-s + (−5.60e7 − 4.31e7i)10-s + 1.27e8·11-s + (7.36e7 − 7.36e7i)12-s + (−1.02e9 − 1.02e9i)13-s + 1.35e9i·14-s + (7.56e8 − 9.83e8i)15-s − 1.07e9·16-s + (−6.56e9 + 6.56e9i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.342 + 0.342i)3-s − 0.5i·4-s + (−0.129 − 0.991i)5-s + 0.342·6-s + (−0.917 + 0.917i)7-s + (−0.250 − 0.250i)8-s − 0.765i·9-s + (−0.560 − 0.431i)10-s + 0.595·11-s + (0.171 − 0.171i)12-s + (−1.25 − 1.25i)13-s + 0.917i·14-s + (0.295 − 0.383i)15-s − 0.250·16-s + (−0.941 + 0.941i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.935 + 0.353i$
Analytic conductor: \(16.2324\)
Root analytic conductor: \(4.02895\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :8),\ -0.935 + 0.353i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.247046 - 1.35133i\)
\(L(\frac12)\) \(\approx\) \(0.247046 - 1.35133i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-128 + 128i)T \)
5 \( 1 + (5.05e4 + 3.87e5i)T \)
good3 \( 1 + (-2.24e3 - 2.24e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (5.28e6 - 5.28e6i)T - 3.32e13iT^{2} \)
11 \( 1 - 1.27e8T + 4.59e16T^{2} \)
13 \( 1 + (1.02e9 + 1.02e9i)T + 6.65e17iT^{2} \)
17 \( 1 + (6.56e9 - 6.56e9i)T - 4.86e19iT^{2} \)
19 \( 1 + 7.99e9iT - 2.88e20T^{2} \)
23 \( 1 + (8.23e10 + 8.23e10i)T + 6.13e21iT^{2} \)
29 \( 1 - 1.95e11iT - 2.50e23T^{2} \)
31 \( 1 - 1.11e12T + 7.27e23T^{2} \)
37 \( 1 + (-3.13e12 + 3.13e12i)T - 1.23e25iT^{2} \)
41 \( 1 - 9.50e11T + 6.37e25T^{2} \)
43 \( 1 + (-8.52e12 - 8.52e12i)T + 1.36e26iT^{2} \)
47 \( 1 + (-1.33e13 + 1.33e13i)T - 5.66e26iT^{2} \)
53 \( 1 + (-2.22e12 - 2.22e12i)T + 3.87e27iT^{2} \)
59 \( 1 - 5.45e13iT - 2.15e28T^{2} \)
61 \( 1 - 2.17e14T + 3.67e28T^{2} \)
67 \( 1 + (7.25e13 - 7.25e13i)T - 1.64e29iT^{2} \)
71 \( 1 - 3.26e13T + 4.16e29T^{2} \)
73 \( 1 + (8.08e14 + 8.08e14i)T + 6.50e29iT^{2} \)
79 \( 1 - 8.90e14iT - 2.30e30T^{2} \)
83 \( 1 + (2.53e15 + 2.53e15i)T + 5.07e30iT^{2} \)
89 \( 1 + 1.01e15iT - 1.54e31T^{2} \)
97 \( 1 + (-2.16e15 + 2.16e15i)T - 6.14e31iT^{2} \)
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

−15.82500360825657559871008570944, −14.83334017318852693053463215798, −12.83801121121617977498079940513, −12.15531122237981836032966619105, −9.889072376756020082131328293505, −8.755110485963084842088953446247, −6.03769748486652158624918253672, −4.28557847688585406812305623710, −2.63614155904710481153393648881, −0.41208819966126437659461025270, 2.48497177222837716478224938128, 4.17368924875146027802303237824, 6.62238180794697822979622506422, 7.48892499895097391974262788730, 9.811745666951658025405085109046, 11.68384869787854985844820553041, 13.58720633830927078888685976548, 14.24588580745959539545094319354, 15.94232931107162404896495484498, 17.20313562762980628183085522451

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