↑

Properties

Label 2-5-5.3-c18-0-6
Degree $2$
Conductor $5$
Sign $0.0217 + 0.999i$
Analytic cond. $10.2693$
Root an. cond. $3.20457$
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  + (514. − 514. i)2-s + (1.17e4 + 1.17e4i)3-s − 2.67e5i·4-s + (−1.06e6 − 1.63e6i)5-s + 1.20e7·6-s + (4.54e7 − 4.54e7i)7-s + (−2.58e6 − 2.58e6i)8-s − 1.13e8i·9-s + (−1.38e9 − 2.96e8i)10-s + 3.09e8·11-s + (3.12e9 − 3.12e9i)12-s + (−8.68e8 − 8.68e8i)13-s − 4.68e10i·14-s + (6.74e9 − 3.16e10i)15-s + 6.73e10·16-s + (−1.22e11 + 1.22e11i)17-s + ⋯
L(s)  = 1  + (1.00 − 1.00i)2-s + (0.594 + 0.594i)3-s − 1.01i·4-s + (−0.544 − 0.839i)5-s + 1.19·6-s + (1.12 − 1.12i)7-s + (−0.0192 − 0.0192i)8-s − 0.292i·9-s + (−1.38 − 0.296i)10-s + 0.131·11-s + (0.606 − 0.606i)12-s + (−0.0818 − 0.0818i)13-s − 2.26i·14-s + (0.175 − 0.822i)15-s + 0.980·16-s + (−1.03 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0217 + 0.999i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.0217 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.0217 + 0.999i$
Analytic conductor: \(10.2693\)
Root analytic conductor: \(3.20457\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :9),\ 0.0217 + 0.999i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(2.49650 - 2.44274i\)
\(L(\frac12)\) \(\approx\) \(2.49650 - 2.44274i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.06e6 + 1.63e6i)T \)
good2 \( 1 + (-514. + 514. i)T - 2.62e5iT^{2} \)
3 \( 1 + (-1.17e4 - 1.17e4i)T + 3.87e8iT^{2} \)
7 \( 1 + (-4.54e7 + 4.54e7i)T - 1.62e15iT^{2} \)
11 \( 1 - 3.09e8T + 5.55e18T^{2} \)
13 \( 1 + (8.68e8 + 8.68e8i)T + 1.12e20iT^{2} \)
17 \( 1 + (1.22e11 - 1.22e11i)T - 1.40e22iT^{2} \)
19 \( 1 - 3.40e11iT - 1.04e23T^{2} \)
23 \( 1 + (-2.24e10 - 2.24e10i)T + 3.24e24iT^{2} \)
29 \( 1 - 1.17e13iT - 2.10e26T^{2} \)
31 \( 1 - 3.42e13T + 6.99e26T^{2} \)
37 \( 1 + (7.33e13 - 7.33e13i)T - 1.68e28iT^{2} \)
41 \( 1 - 9.80e13T + 1.07e29T^{2} \)
43 \( 1 + (6.23e14 + 6.23e14i)T + 2.52e29iT^{2} \)
47 \( 1 + (6.89e14 - 6.89e14i)T - 1.25e30iT^{2} \)
53 \( 1 + (-4.54e15 - 4.54e15i)T + 1.08e31iT^{2} \)
59 \( 1 - 2.86e15iT - 7.50e31T^{2} \)
61 \( 1 + 1.35e16T + 1.36e32T^{2} \)
67 \( 1 + (-2.72e16 + 2.72e16i)T - 7.40e32iT^{2} \)
71 \( 1 + 1.52e16T + 2.10e33T^{2} \)
73 \( 1 + (-8.25e15 - 8.25e15i)T + 3.46e33iT^{2} \)
79 \( 1 + 9.97e16iT - 1.43e34T^{2} \)
83 \( 1 + (-8.47e16 - 8.47e16i)T + 3.49e34iT^{2} \)
89 \( 1 + 7.13e16iT - 1.22e35T^{2} \)
97 \( 1 + (1.73e17 - 1.73e17i)T - 5.77e35iT^{2} \)
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

−19.85901733033918180643369766451, −17.10294263699566528618537382080, −15.02001889673411375854675533203, −13.71821161040174640846795014184, −12.09536001762404716681265978163, −10.56390342838329175383432136549, −8.289481174243173845233360662058, −4.60910658813675089139538548989, −3.75392958326901112711201650627, −1.39463722793724660817379951869, 2.48940728021242223648319385214, 4.84542837944091464175807507407, 6.85734334892394816393229283809, 8.209024911727608375281097436146, 11.57089014741813909405905945004, 13.55085330038935994276491165641, 14.69641346383703963619965223377, 15.65994760413017288912991524809, 18.05484873403953673919491685159, 19.43991786534593654425179386037

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