↑

Properties

Label 2-210-35.12-c1-0-7
Degree $2$
Conductor $210$
Sign $0.628 + 0.777i$
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  + (0.965 − 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.499i)4-s + (1.55 − 1.61i)5-s − i·6-s + (−1.38 + 2.25i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.08 − 1.95i)10-s + (0.582 + 1.00i)11-s + (−0.258 − 0.965i)12-s + (−1.92 − 1.92i)13-s + (−0.756 + 2.53i)14-s + (−1.15 − 1.91i)15-s + (0.500 − 0.866i)16-s + (−0.0209 − 0.00560i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.249i)4-s + (0.693 − 0.720i)5-s − 0.408i·6-s + (−0.524 + 0.851i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.341 − 0.618i)10-s + (0.175 + 0.304i)11-s + (−0.0747 − 0.278i)12-s + (−0.533 − 0.533i)13-s + (−0.202 + 0.677i)14-s + (−0.298 − 0.494i)15-s + (0.125 − 0.216i)16-s + (−0.00507 − 0.00136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.628 + 0.777i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.628 + 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69054 - 0.807613i\)
\(L(\frac12)\) \(\approx\) \(1.69054 - 0.807613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-1.55 + 1.61i)T \)
7 \( 1 + (1.38 - 2.25i)T \)
good11 \( 1 + (-0.582 - 1.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.92 + 1.92i)T + 13iT^{2} \)
17 \( 1 + (0.0209 + 0.00560i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.989 - 1.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.85 - 6.93i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.60iT - 29T^{2} \)
31 \( 1 + (-6.86 + 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (10.2 - 2.74i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.48iT - 41T^{2} \)
43 \( 1 + (7.87 - 7.87i)T - 43iT^{2} \)
47 \( 1 + (1.05 + 3.94i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.82 - 0.757i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.34 - 9.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.15 + 1.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.76 + 14.0i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.51T + 71T^{2} \)
73 \( 1 + (-0.969 + 3.61i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.39 + 0.805i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.74 + 9.74i)T + 83iT^{2} \)
89 \( 1 + (-1.80 + 3.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.265 - 0.265i)T - 97iT^{2} \)
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

−12.37318289980475665879007120223, −11.75767495984211451383110584054, −10.20030085211047380719584911519, −9.349548160185982195062605827633, −8.274980624013581445757433302812, −6.88523941148322900271997244728, −5.82299536761653644859280524314, −4.98704420479670155194853037240, −3.16561949050084236796912819948, −1.79485023929811537208191051065, 2.58128882892844727584666324334, 3.80790977953251151354013237807, 5.00498983295720911561475008730, 6.42893678217930556832848627831, 7.01488732137256503563483475314, 8.570792031456780329175306723966, 9.873201904438100315121964087448, 10.48228273542813756989519567157, 11.47393535835881726280186725998, 12.70288707566503090606552962892

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