↑

Properties

Label 2-546-91.73-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.219 + 0.975i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)3-s + (−0.866 − 0.499i)4-s + (2.96 + 0.794i)5-s + (−0.707 + 0.707i)6-s + (−2.29 + 1.32i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (1.53 − 2.66i)10-s + (−1.52 − 5.70i)11-s + (0.5 + 0.866i)12-s + (2.68 − 2.40i)13-s + (0.684 + 2.55i)14-s + (−2.17 − 2.17i)15-s + (0.500 + 0.866i)16-s + (2.42 − 4.19i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.499 − 0.288i)3-s + (−0.433 − 0.249i)4-s + (1.32 + 0.355i)5-s + (−0.288 + 0.288i)6-s + (−0.866 + 0.499i)7-s + (−0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (0.485 − 0.841i)10-s + (−0.460 − 1.71i)11-s + (0.144 + 0.249i)12-s + (0.744 − 0.667i)13-s + (0.182 + 0.683i)14-s + (−0.560 − 0.560i)15-s + (0.125 + 0.216i)16-s + (0.587 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.219 + 0.975i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.219 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893174 - 1.11648i\)
\(L(\frac12)\) \(\approx\) \(0.893174 - 1.11648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (2.29 - 1.32i)T \)
13 \( 1 + (-2.68 + 2.40i)T \)
good5 \( 1 + (-2.96 - 0.794i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.52 + 5.70i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.42 + 4.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.47 - 0.663i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.88 + 1.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.52T + 29T^{2} \)
31 \( 1 + (2.13 + 7.96i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (6.74 + 1.80i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.208 - 0.208i)T - 41iT^{2} \)
43 \( 1 - 4.79iT - 43T^{2} \)
47 \( 1 + (2.11 - 7.89i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.36 - 5.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.22 - 1.39i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.00 - 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.0 + 3.49i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.97 - 7.97i)T + 71iT^{2} \)
73 \( 1 + (-15.5 + 4.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.82 + 6.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.84 - 5.84i)T - 83iT^{2} \)
89 \( 1 + (-0.865 + 3.23i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.75 - 5.75i)T - 97iT^{2} \)
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

−10.65615936584208377953130212676, −9.812616535778936272350994104118, −9.111930784214736568469292732571, −7.985147836918772318372189569408, −6.44620062229335537846168752809, −5.85811424188644356182918707202, −5.26701877354691653316262961934, −3.29196778825969023537298782383, −2.63500767885642996635783531636, −0.897689421508459036577810904111, 1.65365701650386610488990106602, 3.52085831810639425994088899854, 4.78004306656053093194879029300, 5.49312426461409989980825464941, 6.56501252196262733875338527945, 7.00810408196603930921720774390, 8.463124129628543279401374151090, 9.542521505431231999560786991100, 9.945446411704539992988798182618, 10.71878618663336087182407688238

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