↑

Properties

Label 2-105-3.2-c2-0-14
Degree $2$
Conductor $105$
Sign $-0.871 - 0.490i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
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  − 3.73i·2-s + (1.47 − 2.61i)3-s − 9.96·4-s + 2.23i·5-s + (−9.77 − 5.49i)6-s − 2.64·7-s + 22.2i·8-s + (−4.67 − 7.68i)9-s + 8.35·10-s − 13.0i·11-s + (−14.6 + 26.0i)12-s + 10.2·13-s + 9.88i·14-s + (5.84 + 3.28i)15-s + 43.4·16-s − 21.6i·17-s + ⋯
L(s)  = 1  − 1.86i·2-s + (0.490 − 0.871i)3-s − 2.49·4-s + 0.447i·5-s + (−1.62 − 0.915i)6-s − 0.377·7-s + 2.78i·8-s + (−0.519 − 0.854i)9-s + 0.835·10-s − 1.19i·11-s + (−1.22 + 2.17i)12-s + 0.789·13-s + 0.706i·14-s + (0.389 + 0.219i)15-s + 2.71·16-s − 1.27i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.871 - 0.490i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.871 - 0.490i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.307492 + 1.17426i\)
\(L(\frac12)\) \(\approx\) \(0.307492 + 1.17426i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 + 2.61i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 + 2.64T \)
good2 \( 1 + 3.73iT - 4T^{2} \)
11 \( 1 + 13.0iT - 121T^{2} \)
13 \( 1 - 10.2T + 169T^{2} \)
17 \( 1 + 21.6iT - 289T^{2} \)
19 \( 1 - 21.6T + 361T^{2} \)
23 \( 1 - 33.9iT - 529T^{2} \)
29 \( 1 + 23.2iT - 841T^{2} \)
31 \( 1 + 7.46T + 961T^{2} \)
37 \( 1 - 29.2T + 1.36e3T^{2} \)
41 \( 1 + 6.98iT - 1.68e3T^{2} \)
43 \( 1 - 51.1T + 1.84e3T^{2} \)
47 \( 1 - 43.1iT - 2.20e3T^{2} \)
53 \( 1 - 16.4iT - 2.80e3T^{2} \)
59 \( 1 + 32.7iT - 3.48e3T^{2} \)
61 \( 1 + 15.5T + 3.72e3T^{2} \)
67 \( 1 - 34.1T + 4.48e3T^{2} \)
71 \( 1 - 127. iT - 5.04e3T^{2} \)
73 \( 1 + 78.3T + 5.32e3T^{2} \)
79 \( 1 - 67.0T + 6.24e3T^{2} \)
83 \( 1 - 81.5iT - 6.88e3T^{2} \)
89 \( 1 + 139. iT - 7.92e3T^{2} \)
97 \( 1 - 128.T + 9.40e3T^{2} \)
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

−12.88329737451528298120758634637, −11.59993287773423427464300145432, −11.27143624754541724862409444958, −9.727847526120320998224170727562, −8.964912883547256572899530047906, −7.64568370433557167164378531925, −5.78318407157289589211498803921, −3.59316776669071923901502176606, −2.76039994968685932572730784999, −0.954657516854896553730999586824, 3.94144592254180022024689747778, 4.97752853207106223097553796063, 6.19768668689129612946401733408, 7.57044908269757563812847031427, 8.585215830265753778794778774237, 9.366118585777594680247607159466, 10.41206543023200369072815861940, 12.58041963552804836802573036473, 13.51973456653383521621499822417, 14.57731134092183829898742886585

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