↑

Properties

Label 2-210-105.47-c2-0-7
Degree $2$
Conductor $210$
Sign $0.949 - 0.314i$
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  + (−0.366 − 1.36i)2-s + (−2.86 + 0.897i)3-s + (−1.73 + i)4-s + (−4.37 − 2.42i)5-s + (2.27 + 3.58i)6-s + (−6.90 − 1.16i)7-s + (2 + 1.99i)8-s + (7.39 − 5.13i)9-s + (−1.71 + 6.86i)10-s + (7.83 − 4.52i)11-s + (4.06 − 4.41i)12-s + (15.3 + 15.3i)13-s + (0.941 + 9.85i)14-s + (14.6 + 3.01i)15-s + (1.99 − 3.46i)16-s + (−6.74 + 25.1i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.954 + 0.299i)3-s + (−0.433 + 0.250i)4-s + (−0.874 − 0.484i)5-s + (0.378 + 0.597i)6-s + (−0.986 − 0.165i)7-s + (0.250 + 0.249i)8-s + (0.821 − 0.570i)9-s + (−0.171 + 0.686i)10-s + (0.712 − 0.411i)11-s + (0.338 − 0.368i)12-s + (1.18 + 1.18i)13-s + (0.0672 + 0.703i)14-s + (0.979 + 0.201i)15-s + (0.124 − 0.216i)16-s + (−0.396 + 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.949 - 0.314i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.628545 + 0.101305i\)
\(L(\frac12)\) \(\approx\) \(0.628545 + 0.101305i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
3 \( 1 + (2.86 - 0.897i)T \)
5 \( 1 + (4.37 + 2.42i)T \)
7 \( 1 + (6.90 + 1.16i)T \)
good11 \( 1 + (-7.83 + 4.52i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-15.3 - 15.3i)T + 169iT^{2} \)
17 \( 1 + (6.74 - 25.1i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (-0.500 + 0.867i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.7 + 3.42i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 8.69T + 841T^{2} \)
31 \( 1 + (-29.5 + 17.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (16.9 - 4.54i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 18.4T + 1.68e3T^{2} \)
43 \( 1 + (-24.0 + 24.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-30.4 + 8.15i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (12.8 - 47.9i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (78.7 - 45.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-56.9 - 32.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-3.51 + 13.1i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 86.5iT - 5.04e3T^{2} \)
73 \( 1 + (26.9 - 100. i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-69.3 - 40.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (11.6 - 11.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-62.2 - 35.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (5.41 - 5.41i)T - 9.40e3iT^{2} \)
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

−12.01832232486196438034329786802, −11.25702470475293553010549485406, −10.52273005878869441437932816993, −9.282549325175172345427060266247, −8.578337204615373328664868581704, −6.90942534376350514344978802664, −5.97962320227825852636536293229, −4.24633915972587284498465739574, −3.72761488043723967487101887015, −1.10115589097198661141321106076, 0.55599226651988016009219231946, 3.37442008801225251004619157166, 4.84149256273508252269945073451, 6.16686191505640405422511660917, 6.85236417657143067222692138801, 7.74887934687781412845155973241, 9.065951155385261805443547516925, 10.22484955663218801689961826339, 11.14516535752689448779471337561, 12.07860276147969291920328722036

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