Properties

Label 2-15-15.2-c9-0-10
Degree $2$
Conductor $15$
Sign $-0.166 + 0.986i$
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  + (−6.35 + 6.35i)2-s + (−140. − 0.342i)3-s + 431. i·4-s + (1.05e3 + 919. i)5-s + (893. − 889. i)6-s + (−7.86e3 − 7.86e3i)7-s + (−5.99e3 − 5.99e3i)8-s + (1.96e4 + 96.0i)9-s + (−1.25e4 + 839. i)10-s − 5.07e4i·11-s + (147. − 6.04e4i)12-s + (−4.94e3 + 4.94e3i)13-s + 9.99e4·14-s + (−1.47e5 − 1.29e5i)15-s − 1.44e5·16-s + (1.74e5 − 1.74e5i)17-s + ⋯
L(s)  = 1  + (−0.280 + 0.280i)2-s + (−0.999 − 0.00243i)3-s + 0.842i·4-s + (0.752 + 0.658i)5-s + (0.281 − 0.280i)6-s + (−1.23 − 1.23i)7-s + (−0.517 − 0.517i)8-s + (0.999 + 0.00487i)9-s + (−0.396 + 0.0265i)10-s − 1.04i·11-s + (0.00205 − 0.842i)12-s + (−0.0480 + 0.0480i)13-s + 0.695·14-s + (−0.751 − 0.660i)15-s − 0.551·16-s + (0.507 − 0.507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(15\)    =    \(3 \cdot 5\)
Sign: $-0.166 + 0.986i$
Analytic conductor: \(7.72553\)
Root analytic conductor: \(2.77948\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{15} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 15,\ (\ :9/2),\ -0.166 + 0.986i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.224983 - 0.266176i\)
\(L(\frac12)\) \(\approx\) \(0.224983 - 0.266176i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (140. + 0.342i)T \)
5 \( 1 + (-1.05e3 - 919. i)T \)
good2 \( 1 + (6.35 - 6.35i)T - 512iT^{2} \)
7 \( 1 + (7.86e3 + 7.86e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 5.07e4iT - 2.35e9T^{2} \)
13 \( 1 + (4.94e3 - 4.94e3i)T - 1.06e10iT^{2} \)
17 \( 1 + (-1.74e5 + 1.74e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 3.60e4iT - 3.22e11T^{2} \)
23 \( 1 + (9.95e5 + 9.95e5i)T + 1.80e12iT^{2} \)
29 \( 1 + 1.37e6T + 1.45e13T^{2} \)
31 \( 1 + 5.36e6T + 2.64e13T^{2} \)
37 \( 1 + (4.18e6 + 4.18e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.69e7iT - 3.27e14T^{2} \)
43 \( 1 + (-4.04e6 + 4.04e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (3.93e7 - 3.93e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (4.34e7 + 4.34e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 1.26e8T + 8.66e15T^{2} \)
61 \( 1 + 1.56e8T + 1.16e16T^{2} \)
67 \( 1 + (7.00e7 + 7.00e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 3.78e7iT - 4.58e16T^{2} \)
73 \( 1 + (-1.07e8 + 1.07e8i)T - 5.88e16iT^{2} \)
79 \( 1 - 6.60e7iT - 1.19e17T^{2} \)
83 \( 1 + (-3.05e8 - 3.05e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 4.49e7T + 3.50e17T^{2} \)
97 \( 1 + (1.77e8 + 1.77e8i)T + 7.60e17iT^{2} \)
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

−16.69924376910559785529676440010, −16.20034037799159283563974055326, −13.79073183254233479226081432043, −12.67204768065423243394151261362, −10.92321286411559377859671107609, −9.679997859180825897423152639575, −7.25328366448422053722713781448, −6.19897463264964757868351767423, −3.51961274540368439748814843124, −0.21878069825677612834486054859, 1.77384377851754170975368107878, 5.30582271681316837599923884648, 6.26603313926811457383391336677, 9.366468637233027801087140548774, 10.08851583272041557913659803019, 11.98276674065658459821723439720, 12.99196245888585125644215460779, 15.11752544015287374533848482143, 16.29464184563866636618734794506, 17.73242528272606036016650741983

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