↑

Properties

Label 2-210-105.17-c2-0-10
Degree $2$
Conductor $210$
Sign $0.966 + 0.258i$
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  + (−1.36 − 0.366i)2-s + (−1.37 − 2.66i)3-s + (1.73 + i)4-s + (4.92 − 0.860i)5-s + (0.898 + 4.14i)6-s + (1.68 + 6.79i)7-s + (−1.99 − 2i)8-s + (−5.23 + 7.32i)9-s + (−7.04 − 0.627i)10-s + (13.7 + 7.94i)11-s + (0.290 − 5.99i)12-s + (5.32 + 5.32i)13-s + (0.185 − 9.89i)14-s + (−9.05 − 11.9i)15-s + (1.99 + 3.46i)16-s + (−16.9 + 4.55i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.457 − 0.889i)3-s + (0.433 + 0.250i)4-s + (0.985 − 0.172i)5-s + (0.149 + 0.691i)6-s + (0.240 + 0.970i)7-s + (−0.249 − 0.250i)8-s + (−0.581 + 0.813i)9-s + (−0.704 − 0.0627i)10-s + (1.25 + 0.722i)11-s + (0.0242 − 0.499i)12-s + (0.409 + 0.409i)13-s + (0.0132 − 0.706i)14-s + (−0.603 − 0.797i)15-s + (0.124 + 0.216i)16-s + (−0.999 + 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.966 + 0.258i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.966 + 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20000 - 0.157584i\)
\(L(\frac12)\) \(\approx\) \(1.20000 - 0.157584i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + (1.37 + 2.66i)T \)
5 \( 1 + (-4.92 + 0.860i)T \)
7 \( 1 + (-1.68 - 6.79i)T \)
good11 \( 1 + (-13.7 - 7.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5.32 - 5.32i)T + 169iT^{2} \)
17 \( 1 + (16.9 - 4.55i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-5.62 - 9.74i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.4 + 39.1i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 22.8T + 841T^{2} \)
31 \( 1 + (-36.0 - 20.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-7.72 + 28.8i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 55.0T + 1.68e3T^{2} \)
43 \( 1 + (14.2 - 14.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-6.77 + 25.2i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (59.0 - 15.8i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (10.7 + 6.18i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-14.5 + 8.40i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-103. + 27.8i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 34.7iT - 5.04e3T^{2} \)
73 \( 1 + (56.8 - 15.2i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-40.5 + 23.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (75.6 - 75.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-116. + 67.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (75.7 - 75.7i)T - 9.40e3iT^{2} \)
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

−12.14088435520827925901302049394, −11.22542411693013961481949721769, −10.11896698461476316587002907635, −8.961872481128609776997922535533, −8.403439964409219244886087659451, −6.66823107075142285794970025774, −6.34852944345967295554670939692, −4.84475846991778211926515987943, −2.40392979323122273553989861532, −1.42152415285904326000625167906, 1.08565339690866557362146119812, 3.33361758592464604307176709983, 4.83127748996291018710977346772, 6.11829408959282641097967933430, 6.86476836016899686199951298806, 8.498420803069401611090834976853, 9.425303809624145887190497600884, 10.09511045385224707175932086417, 11.10616640182350123397396129724, 11.58202017186650773168386662331

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