↑

Properties

Label 2-210-105.17-c2-0-8
Degree $2$
Conductor $210$
Sign $0.713 - 0.700i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (−2.98 − 0.278i)3-s + (1.73 + i)4-s + (−4.38 − 2.39i)5-s + (−3.97 − 1.47i)6-s + (5.69 + 4.07i)7-s + (1.99 + 2i)8-s + (8.84 + 1.66i)9-s + (−5.11 − 4.87i)10-s + (12.1 + 7.01i)11-s + (−4.89 − 3.46i)12-s + (7.32 + 7.32i)13-s + (6.28 + 7.64i)14-s + (12.4 + 8.37i)15-s + (1.99 + 3.46i)16-s + (−8.77 + 2.35i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.995 − 0.0929i)3-s + (0.433 + 0.250i)4-s + (−0.877 − 0.478i)5-s + (−0.663 − 0.245i)6-s + (0.813 + 0.581i)7-s + (0.249 + 0.250i)8-s + (0.982 + 0.185i)9-s + (−0.511 − 0.487i)10-s + (1.10 + 0.637i)11-s + (−0.407 − 0.289i)12-s + (0.563 + 0.563i)13-s + (0.449 + 0.546i)14-s + (0.829 + 0.558i)15-s + (0.124 + 0.216i)16-s + (−0.515 + 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.54373 + 0.631103i\)
\(L(\frac12)\) \(\approx\) \(1.54373 + 0.631103i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 + (2.98 + 0.278i)T \)
5 \( 1 + (4.38 + 2.39i)T \)
7 \( 1 + (-5.69 - 4.07i)T \)
good11 \( 1 + (-12.1 - 7.01i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.32 - 7.32i)T + 169iT^{2} \)
17 \( 1 + (8.77 - 2.35i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-13.9 - 24.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.52 + 31.8i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 4.58T + 841T^{2} \)
31 \( 1 + (31.4 + 18.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (9.81 - 36.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 53.2T + 1.68e3T^{2} \)
43 \( 1 + (-16.6 + 16.6i)T - 1.84e3iT^{2} \)
47 \( 1 + (-4.62 + 17.2i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (78.9 - 21.1i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (8.07 + 4.66i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-92.7 + 53.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (90.5 - 24.2i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 31.9iT - 5.04e3T^{2} \)
73 \( 1 + (63.8 - 17.1i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-46.9 + 27.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-37.9 + 37.9i)T - 6.88e3iT^{2} \)
89 \( 1 + (91.9 - 53.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-59.6 + 59.6i)T - 9.40e3iT^{2} \)
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

−12.14225077326809143792145862958, −11.61663041999482725507482793137, −10.81381010030281107028528855071, −9.217053362515389192614240798542, −8.051523173928282364131267585740, −6.96283634992336601858690856448, −5.91538261645077629305763156183, −4.72614679815147414536441935947, −4.01746781451921427359007476348, −1.55000878552850389316452901773, 1.02298856599883700584060838015, 3.48341277544907169180631715823, 4.43517170883746491268050900530, 5.59085087607622791777117881087, 6.83154040617247620392920139474, 7.57302148854172662835958405832, 9.171514217537509782707008078382, 10.75012904294226295935944791144, 11.22580997408757233708496438042, 11.67509300636259538060302439209

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