↑

Properties

Label 2-2016-168.101-c1-0-15
Degree $2$
Conductor $2016$
Sign $0.995 + 0.0935i$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.16 + 1.82i)5-s + (1.64 − 2.06i)7-s + (−0.200 + 0.346i)11-s + 0.581·13-s + (−2.27 + 3.94i)17-s + (−2.43 − 4.21i)19-s + (6.21 − 3.58i)23-s + (4.15 − 7.20i)25-s − 6.69·29-s + (2.85 + 1.64i)31-s + (−1.43 + 9.54i)35-s + (1.93 − 1.11i)37-s + 4.51·41-s + 4.09i·43-s + (2.86 + 4.96i)47-s + ⋯
L(s)  = 1  + (−1.41 + 0.815i)5-s + (0.623 − 0.782i)7-s + (−0.0603 + 0.104i)11-s + 0.161·13-s + (−0.552 + 0.956i)17-s + (−0.557 − 0.966i)19-s + (1.29 − 0.747i)23-s + (0.831 − 1.44i)25-s − 1.24·29-s + (0.513 + 0.296i)31-s + (−0.242 + 1.61i)35-s + (0.317 − 0.183i)37-s + 0.705·41-s + 0.624i·43-s + (0.417 + 0.723i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.995 + 0.0935i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ 0.995 + 0.0935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.270866842\)
\(L(\frac12)\) \(\approx\) \(1.270866842\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-1.64 + 2.06i)T \)
good5 \( 1 + (3.16 - 1.82i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.200 - 0.346i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.581T + 13T^{2} \)
17 \( 1 + (2.27 - 3.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.43 + 4.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.21 + 3.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.69T + 29T^{2} \)
31 \( 1 + (-2.85 - 1.64i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.93 + 1.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.51T + 41T^{2} \)
43 \( 1 - 4.09iT - 43T^{2} \)
47 \( 1 + (-2.86 - 4.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.74 + 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.72 - 4.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.42 - 11.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.30 - 4.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.42iT - 71T^{2} \)
73 \( 1 + (6.18 + 3.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.15 + 7.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (-5.25 - 9.11i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.00iT - 97T^{2} \)
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

−8.833594908483732893208149908752, −8.359974197770665268352429454858, −7.36539237076235255869637749904, −7.11144304233295273449467649270, −6.17190329764982653012046876096, −4.78420331457464248158643069060, −4.19889406061306151555251236742, −3.43586354149510703054556881509, −2.32169721315585345388312860907, −0.69053201046856056234623177817, 0.812812242631769883437493369924, 2.20233036282716424951620478954, 3.47056093807137045424731917643, 4.29130967428743704314361842140, 5.07083181803609519880550146440, 5.76743554593386728664112624845, 7.07077093694406251968085460535, 7.69615673683925369031633251909, 8.468100496048593943257069842761, 8.893159325751972084172335940726

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