↑

Properties

Label 2-546-91.24-c1-0-5
Degree $2$
Conductor $546$
Sign $0.714 - 0.699i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + i·3-s + (0.866 − 0.499i)4-s + (1.30 + 0.349i)5-s + (−0.258 − 0.965i)6-s + (2.64 + 0.108i)7-s + (−0.707 + 0.707i)8-s − 9-s − 1.35·10-s + (0.402 − 0.402i)11-s + (0.499 + 0.866i)12-s + (3.33 − 1.36i)13-s + (−2.58 + 0.579i)14-s + (−0.349 + 1.30i)15-s + (0.500 − 0.866i)16-s + (0.487 + 0.844i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + 0.577i·3-s + (0.433 − 0.249i)4-s + (0.584 + 0.156i)5-s + (−0.105 − 0.394i)6-s + (0.999 + 0.0410i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s − 0.427·10-s + (0.121 − 0.121i)11-s + (0.144 + 0.249i)12-s + (0.925 − 0.379i)13-s + (−0.689 + 0.154i)14-s + (−0.0903 + 0.337i)15-s + (0.125 − 0.216i)16-s + (0.118 + 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.714 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24407 + 0.507897i\)
\(L(\frac12)\) \(\approx\) \(1.24407 + 0.507897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 - iT \)
7 \( 1 + (-2.64 - 0.108i)T \)
13 \( 1 + (-3.33 + 1.36i)T \)
good5 \( 1 + (-1.30 - 0.349i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.402 + 0.402i)T - 11iT^{2} \)
17 \( 1 + (-0.487 - 0.844i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.29 + 2.29i)T - 19iT^{2} \)
23 \( 1 + (0.0701 + 0.0405i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.139 + 0.242i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.665 - 2.48i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.419 - 1.56i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.23 + 0.599i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.48 - 5.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.48 - 5.55i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.54 - 7.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.84 - 6.90i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 5.72iT - 61T^{2} \)
67 \( 1 + (1.53 + 1.53i)T + 67iT^{2} \)
71 \( 1 + (-0.466 + 0.124i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.81 - 0.753i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.45 - 2.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.22 + 2.22i)T - 83iT^{2} \)
89 \( 1 + (10.2 - 2.73i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.34 + 16.2i)T + (-84.0 + 48.5i)T^{2} \)
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

−10.87059215715413365292006447782, −9.987889280605887529221462892645, −9.153355803867341683445301491446, −8.366058849955755509265690205387, −7.55103498081981153049845476555, −6.24801674463315745167295436017, −5.50567286827907277348118975042, −4.34004407170557328549532706853, −2.86293135308272482723649968368, −1.38759973744403947734166353634, 1.25497567732851316634637105007, 2.17578049299132009178640180930, 3.79933347562352807902938046707, 5.29261883435320709699809547640, 6.20505093639090780867966980771, 7.30009383337139291720121952129, 8.077951801502368783518447728564, 8.885712830588509178915110220245, 9.723528116026302278812301341105, 10.74674081032924810704567476334

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