↑

Properties

Label 2-210-105.62-c2-0-20
Degree $2$
Conductor $210$
Sign $-0.424 + 0.905i$
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 − i)2-s + (−2.09 + 2.14i)3-s − 2i·4-s + (−1.13 − 4.86i)5-s + (0.0523 + 4.24i)6-s + (2.31 + 6.60i)7-s + (−2 − 2i)8-s + (−0.222 − 8.99i)9-s + (−6.00 − 3.73i)10-s − 16.9i·11-s + (4.29 + 4.18i)12-s + (−10.2 − 10.2i)13-s + (8.91 + 4.29i)14-s + (12.8 + 7.76i)15-s − 4·16-s + (−8.79 − 8.79i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.698 + 0.715i)3-s − 0.5i·4-s + (−0.227 − 0.973i)5-s + (0.00872 + 0.707i)6-s + (0.330 + 0.943i)7-s + (−0.250 − 0.250i)8-s + (−0.0246 − 0.999i)9-s + (−0.600 − 0.373i)10-s − 1.54i·11-s + (0.357 + 0.349i)12-s + (−0.786 − 0.786i)13-s + (0.637 + 0.306i)14-s + (0.855 + 0.517i)15-s − 0.250·16-s + (−0.517 − 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.424 + 0.905i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.424 + 0.905i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.640673 - 1.00841i\)
\(L(\frac12)\) \(\approx\) \(0.640673 - 1.00841i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (2.09 - 2.14i)T \)
5 \( 1 + (1.13 + 4.86i)T \)
7 \( 1 + (-2.31 - 6.60i)T \)
good11 \( 1 + 16.9iT - 121T^{2} \)
13 \( 1 + (10.2 + 10.2i)T + 169iT^{2} \)
17 \( 1 + (8.79 + 8.79i)T + 289iT^{2} \)
19 \( 1 - 24.7T + 361T^{2} \)
23 \( 1 + (19.2 + 19.2i)T + 529iT^{2} \)
29 \( 1 + 1.67T + 841T^{2} \)
31 \( 1 + 36.8iT - 961T^{2} \)
37 \( 1 + (-40.5 - 40.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 0.885T + 1.68e3T^{2} \)
43 \( 1 + (9.87 - 9.87i)T - 1.84e3iT^{2} \)
47 \( 1 + (-33.7 - 33.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-11.9 - 11.9i)T + 2.80e3iT^{2} \)
59 \( 1 - 50.5iT - 3.48e3T^{2} \)
61 \( 1 + 80.6iT - 3.72e3T^{2} \)
67 \( 1 + (4.46 + 4.46i)T + 4.48e3iT^{2} \)
71 \( 1 - 137. iT - 5.04e3T^{2} \)
73 \( 1 + (-53.3 - 53.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 127. iT - 6.24e3T^{2} \)
83 \( 1 + (-60.0 + 60.0i)T - 6.88e3iT^{2} \)
89 \( 1 + 51.3iT - 7.92e3T^{2} \)
97 \( 1 + (0.274 - 0.274i)T - 9.40e3iT^{2} \)
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

−11.72529874625095904005085855521, −11.25079997146297707574601720812, −9.924984967500737319964466436245, −9.099228484147462632717311293561, −8.046935709362357000900441484125, −5.99888414416486251778154713592, −5.37014679798040229800105560663, −4.41910443353963946952440463845, −2.94362725094596072464369843100, −0.61985528781100270133053909596, 2.05583610812269382976837444511, 4.01061945074904191725534114584, 5.11529828664395222212846015974, 6.55818850437730659166776801538, 7.27702453527297186825022953876, 7.70691229556541325294786923820, 9.718920938250892973325026164507, 10.71325741944502520265586573187, 11.71515220768387790584117391512, 12.34392483022651127531609727056

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