↑

Properties

Label 2-210-35.19-c2-0-7
Degree $2$
Conductor $210$
Sign $0.00461 - 0.999i$
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  + (1.22 + 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s + (1.36 + 4.80i)5-s + 2.44i·6-s + (1.07 − 6.91i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−1.72 + 6.85i)10-s + (8.06 + 13.9i)11-s + (−1.73 + 2.99i)12-s − 2.01·13-s + (6.21 − 7.70i)14-s + (−6.03 + 6.21i)15-s + (−2.00 + 3.46i)16-s + (−1.22 − 2.12i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s + (0.273 + 0.961i)5-s + 0.408i·6-s + (0.153 − 0.988i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.172 + 0.685i)10-s + (0.732 + 1.26i)11-s + (−0.144 + 0.249i)12-s − 0.154·13-s + (0.443 − 0.550i)14-s + (−0.402 + 0.414i)15-s + (−0.125 + 0.216i)16-s + (−0.0722 − 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00461 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.00461 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.00461 - 0.999i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.00461 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73616 + 1.72816i\)
\(L(\frac12)\) \(\approx\) \(1.73616 + 1.72816i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (-1.36 - 4.80i)T \)
7 \( 1 + (-1.07 + 6.91i)T \)
good11 \( 1 + (-8.06 - 13.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.01T + 169T^{2} \)
17 \( 1 + (1.22 + 2.12i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (8.06 + 4.65i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (22.3 + 12.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 37.0T + 841T^{2} \)
31 \( 1 + (-36.9 + 21.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-24.4 - 14.1i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 18.4iT - 1.68e3T^{2} \)
43 \( 1 + 11.7iT - 1.84e3T^{2} \)
47 \( 1 + (-32.0 + 55.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-3.55 + 2.05i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-27.5 + 15.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (80.8 + 46.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-92.7 + 53.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 98.9T + 5.04e3T^{2} \)
73 \( 1 + (27.6 + 47.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (59.3 - 102. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 88.2T + 6.88e3T^{2} \)
89 \( 1 + (97.7 + 56.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 81.0T + 9.40e3T^{2} \)
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

−12.44343846955151259966078904181, −11.41248437466150072236343871145, −10.33351807691457531681842151891, −9.734982576051585589319420871708, −8.166280531501203801996471011236, −7.08721573327633780158292072911, −6.36935345270337523936384255124, −4.67736300106171535768067132902, −3.87003059631717272006576078897, −2.35804863812552814127555857408, 1.27735795001820304967492938051, 2.74494483417218569397471514321, 4.29533189425625513340884630717, 5.66280569519715768333053486307, 6.33354998340074331538234875328, 8.164014818908997666056306157160, 8.807727727378936678690224634870, 9.862316144559332818882507436014, 11.35167167270689472091602946720, 12.09502075079200752755115874950

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