↑

Properties

Label 2-15-5.2-c10-0-0
Degree $2$
Conductor $15$
Sign $0.998 - 0.0536i$
Analytic cond. $9.53035$
Root an. cond. $3.08712$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−19.3 − 19.3i)2-s + (−99.2 + 99.2i)3-s − 273. i·4-s + (−2.99e3 + 880. i)5-s + 3.84e3·6-s + (3.12e3 + 3.12e3i)7-s + (−2.51e4 + 2.51e4i)8-s − 1.96e4i·9-s + (7.51e4 + 4.10e4i)10-s + 2.30e5·11-s + (2.71e4 + 2.71e4i)12-s + (2.87e5 − 2.87e5i)13-s − 1.21e5i·14-s + (2.10e5 − 3.84e5i)15-s + 6.94e5·16-s + (1.39e6 + 1.39e6i)17-s + ⋯
L(s)  = 1  + (−0.605 − 0.605i)2-s + (−0.408 + 0.408i)3-s − 0.266i·4-s + (−0.959 + 0.281i)5-s + 0.494·6-s + (0.186 + 0.186i)7-s + (−0.767 + 0.767i)8-s − 0.333i·9-s + (0.751 + 0.410i)10-s + 1.43·11-s + (0.108 + 0.108i)12-s + (0.774 − 0.774i)13-s − 0.225i·14-s + (0.276 − 0.506i)15-s + 0.661·16-s + (0.979 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.998 - 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $0.998 - 0.0536i$
Analytic conductor: \(9.53035\)
Root analytic conductor: \(3.08712\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :5),\ 0.998 - 0.0536i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.887563 + 0.0238126i\)
\(L(\frac12)\) \(\approx\) \(0.887563 + 0.0238126i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (99.2 - 99.2i)T \)
5 \( 1 + (2.99e3 - 880. i)T \)
good2 \( 1 + (19.3 + 19.3i)T + 1.02e3iT^{2} \)
7 \( 1 + (-3.12e3 - 3.12e3i)T + 2.82e8iT^{2} \)
11 \( 1 - 2.30e5T + 2.59e10T^{2} \)
13 \( 1 + (-2.87e5 + 2.87e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (-1.39e6 - 1.39e6i)T + 2.01e12iT^{2} \)
19 \( 1 - 3.23e5iT - 6.13e12T^{2} \)
23 \( 1 + (7.28e6 - 7.28e6i)T - 4.14e13iT^{2} \)
29 \( 1 + 5.58e6iT - 4.20e14T^{2} \)
31 \( 1 + 3.90e6T + 8.19e14T^{2} \)
37 \( 1 + (-8.90e7 - 8.90e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 3.06e7T + 1.34e16T^{2} \)
43 \( 1 + (-5.99e7 + 5.99e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (-1.25e8 - 1.25e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (2.38e8 - 2.38e8i)T - 1.74e17iT^{2} \)
59 \( 1 - 1.52e8iT - 5.11e17T^{2} \)
61 \( 1 - 9.12e8T + 7.13e17T^{2} \)
67 \( 1 + (-1.02e8 - 1.02e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 3.21e9T + 3.25e18T^{2} \)
73 \( 1 + (1.64e9 - 1.64e9i)T - 4.29e18iT^{2} \)
79 \( 1 - 4.86e9iT - 9.46e18T^{2} \)
83 \( 1 + (-2.32e9 + 2.32e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 7.27e9iT - 3.11e19T^{2} \)
97 \( 1 + (-9.72e8 - 9.72e8i)T + 7.37e19iT^{2} \)
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

−17.11813217106479302048264625456, −15.53126424577790771372670071973, −14.51553407315466156292504744933, −12.04578293262424474636452014219, −11.18150162269918960713747809324, −9.887830916495789865122258339273, −8.289563727962606140857523562452, −5.96743483292252229455453970818, −3.70289452189999495678840454711, −1.10324729685169663535716256847, 0.73849298294793044219788063450, 3.97576762467065547289971943905, 6.54694732822970826290339555728, 7.79517840136688818815951127210, 9.123060169166364318065357437591, 11.49097049028833422809665937582, 12.39536894620685446202103028121, 14.34273361289014141513619066879, 16.15659708380639350765865909036, 16.67467704847182615682969878432

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