↑

Properties

Label 2-210-105.2-c1-0-0
Degree $2$
Conductor $210$
Sign $-0.951 + 0.309i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
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  + (0.258 + 0.965i)2-s + (−0.924 − 1.46i)3-s + (−0.866 + 0.499i)4-s + (−2.13 + 0.675i)5-s + (1.17 − 1.27i)6-s + (−2.45 − 0.998i)7-s + (−0.707 − 0.707i)8-s + (−1.29 + 2.70i)9-s + (−1.20 − 1.88i)10-s + (−5.16 + 2.98i)11-s + (1.53 + 0.806i)12-s + (2.36 − 2.36i)13-s + (0.330 − 2.62i)14-s + (2.95 + 2.49i)15-s + (0.500 − 0.866i)16-s + (0.914 + 0.245i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.533 − 0.845i)3-s + (−0.433 + 0.249i)4-s + (−0.953 + 0.302i)5-s + (0.480 − 0.519i)6-s + (−0.926 − 0.377i)7-s + (−0.249 − 0.249i)8-s + (−0.430 + 0.902i)9-s + (−0.380 − 0.595i)10-s + (−1.55 + 0.899i)11-s + (0.442 + 0.232i)12-s + (0.656 − 0.656i)13-s + (0.0883 − 0.701i)14-s + (0.764 + 0.644i)15-s + (0.125 − 0.216i)16-s + (0.221 + 0.0594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.951 + 0.309i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.951 + 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00158116 - 0.00997835i\)
\(L(\frac12)\) \(\approx\) \(0.00158116 - 0.00997835i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.924 + 1.46i)T \)
5 \( 1 + (2.13 - 0.675i)T \)
7 \( 1 + (2.45 + 0.998i)T \)
good11 \( 1 + (5.16 - 2.98i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.36 + 2.36i)T - 13iT^{2} \)
17 \( 1 + (-0.914 - 0.245i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.22 + 2.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.208 - 0.0558i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 + (-1.28 - 2.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.31 + 0.352i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.15iT - 41T^{2} \)
43 \( 1 + (6.02 - 6.02i)T - 43iT^{2} \)
47 \( 1 + (0.969 + 3.61i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.60 - 5.98i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.02 + 3.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.70 - 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.48 + 9.28i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (12.8 + 3.43i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.874 + 0.504i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.84 + 7.84i)T + 83iT^{2} \)
89 \( 1 + (0.188 - 0.327i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.42 - 3.42i)T + 97iT^{2} \)
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

−12.92786420311419080623804222380, −12.29475271773297560358410511404, −10.95413039888743656357737259816, −10.21565109212758991284882677934, −8.471385395261703525975408026246, −7.59895735233867421779477533144, −6.94231470736646346421690148188, −5.86107101639502261208791791735, −4.57406933461418764117682300429, −2.96107523285610599821120547227, 0.008237845017010287353987492712, 3.08047835868228179356616563637, 4.03300396916165304497833559026, 5.27728339901409980106832604601, 6.31179426633495407538236082740, 8.152996583499059437562422081077, 9.021038379790322932566541317779, 10.14888299372123515346399902159, 10.94098359440141355738999050935, 11.69946883238365868670161189800

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