↑

Properties

Label 2-15-15.8-c9-0-6
Degree $2$
Conductor $15$
Sign $-0.778 - 0.627i$
Analytic cond. $7.72553$
Root an. cond. $2.77948$
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  + (24.0 + 24.0i)2-s + (−17.1 + 139. i)3-s + 647. i·4-s + (1.39e3 + 2.62i)5-s + (−3.76e3 + 2.93e3i)6-s + (−4.15e3 + 4.15e3i)7-s + (−3.27e3 + 3.27e3i)8-s + (−1.90e4 − 4.78e3i)9-s + (3.35e4 + 3.37e4i)10-s − 5.33e4i·11-s + (−9.02e4 − 1.11e4i)12-s + (2.32e4 + 2.32e4i)13-s − 2.00e5·14-s + (−2.43e4 + 1.94e5i)15-s + 1.74e5·16-s + (2.52e5 + 2.52e5i)17-s + ⋯
L(s)  = 1  + (1.06 + 1.06i)2-s + (−0.122 + 0.992i)3-s + 1.26i·4-s + (0.999 + 0.00187i)5-s + (−1.18 + 0.925i)6-s + (−0.653 + 0.653i)7-s + (−0.282 + 0.282i)8-s + (−0.969 − 0.243i)9-s + (1.06 + 1.06i)10-s − 1.09i·11-s + (−1.25 − 0.154i)12-s + (0.226 + 0.226i)13-s − 1.39·14-s + (−0.124 + 0.992i)15-s + 0.664·16-s + (0.732 + 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.778 - 0.627i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -0.778 - 0.627i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.973264 + 2.75901i\)
\(L(\frac12)\) \(\approx\) \(0.973264 + 2.75901i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (17.1 - 139. i)T \)
5 \( 1 + (-1.39e3 - 2.62i)T \)
good2 \( 1 + (-24.0 - 24.0i)T + 512iT^{2} \)
7 \( 1 + (4.15e3 - 4.15e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 5.33e4iT - 2.35e9T^{2} \)
13 \( 1 + (-2.32e4 - 2.32e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-2.52e5 - 2.52e5i)T + 1.18e11iT^{2} \)
19 \( 1 - 5.88e5iT - 3.22e11T^{2} \)
23 \( 1 + (-1.34e6 + 1.34e6i)T - 1.80e12iT^{2} \)
29 \( 1 - 1.67e6T + 1.45e13T^{2} \)
31 \( 1 + 9.03e6T + 2.64e13T^{2} \)
37 \( 1 + (-1.99e6 + 1.99e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 1.85e7iT - 3.27e14T^{2} \)
43 \( 1 + (2.46e7 + 2.46e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (-2.25e6 - 2.25e6i)T + 1.11e15iT^{2} \)
53 \( 1 + (6.59e6 - 6.59e6i)T - 3.29e15iT^{2} \)
59 \( 1 + 2.19e7T + 8.66e15T^{2} \)
61 \( 1 - 9.47e7T + 1.16e16T^{2} \)
67 \( 1 + (2.46e6 - 2.46e6i)T - 2.72e16iT^{2} \)
71 \( 1 + 3.48e8iT - 4.58e16T^{2} \)
73 \( 1 + (1.28e8 + 1.28e8i)T + 5.88e16iT^{2} \)
79 \( 1 + 2.62e8iT - 1.19e17T^{2} \)
83 \( 1 + (1.32e8 - 1.32e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 9.05e8T + 3.50e17T^{2} \)
97 \( 1 + (-4.92e8 + 4.92e8i)T - 7.60e17iT^{2} \)
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

−16.84250993311939769912611334480, −16.29598418521968314908342100049, −14.92235635410668041002044421054, −14.01381739455963759948153401180, −12.61071528519117889362437850570, −10.43455453079445500314403053406, −8.820271413473357917824212155579, −6.20532059332618200868024439536, −5.42593478343341418818616935368, −3.44689314109613395005801789729, 1.33659084327527952575280201138, 2.89621731062119832332513321158, 5.25116419149885791226783428357, 7.02372146048276482239956223922, 9.781516159415407393959197731745, 11.31372746663360961736725718361, 12.84874225197383109715601433437, 13.31027769820727116842845061688, 14.51429874355851315817105352406, 16.95252480285624794789882474487

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