Properties

Label 2-210-35.33-c1-0-7
Degree $2$
Conductor $210$
Sign $0.423 + 0.905i$
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 + (0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s + (1.96 − 1.07i)5-s i·6-s + (−1.52 − 2.15i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−1.54 − 1.61i)10-s + (0.883 + 1.52i)11-s + (−0.965 + 0.258i)12-s + (2.71 − 2.71i)13-s + (−1.69 + 2.03i)14-s + (2.17 − 0.531i)15-s + (0.500 − 0.866i)16-s + (−0.574 + 2.14i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.876 − 0.480i)5-s − 0.408i·6-s + (−0.577 − 0.816i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.488 − 0.510i)10-s + (0.266 + 0.461i)11-s + (−0.278 + 0.0747i)12-s + (0.752 − 0.752i)13-s + (−0.451 + 0.543i)14-s + (0.560 − 0.137i)15-s + (0.125 − 0.216i)16-s + (−0.139 + 0.520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.423 + 0.905i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.423 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15692 - 0.736195i\)
\(L(\frac12)\) \(\approx\) \(1.15692 - 0.736195i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (-1.96 + 1.07i)T \)
7 \( 1 + (1.52 + 2.15i)T \)
good11 \( 1 + (-0.883 - 1.52i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.71 + 2.71i)T - 13iT^{2} \)
17 \( 1 + (0.574 - 2.14i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.886 + 1.53i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.90 - 1.04i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.84iT - 29T^{2} \)
31 \( 1 + (8.94 - 5.16i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.861 + 3.21i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + (-3.46 - 3.46i)T + 43iT^{2} \)
47 \( 1 + (-5.93 + 1.59i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.106 + 0.396i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.18 - 8.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.87 + 3.39i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.37 + 1.97i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (-10.2 - 2.75i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (10.9 + 6.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.94 - 1.94i)T - 83iT^{2} \)
89 \( 1 + (0.558 - 0.966i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.26 + 7.26i)T + 97iT^{2} \)
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

−12.49320565952771124997643230151, −10.92054954362276286285981468836, −10.19063231052130344564668818355, −9.389788211921388740355775414626, −8.541577232666635067015705082384, −7.26929716601618778183227677658, −5.85439049924722533734761666574, −4.37196415473425852511301767210, −3.18210304777575811407938310995, −1.50992291020948867696750519015, 2.16827140686603852223399571930, 3.73906282589903083937683315606, 5.65189647578177810448058018737, 6.32884405002945171716543360932, 7.40746710895612195176442180502, 8.816096784473200067353091663394, 9.263987510609683967870982362408, 10.29471938737382343159517970106, 11.59639904886651443655256864958, 12.83388027702011954595230832713

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