Properties

Label 2-6840-57.56-c1-0-57
Degree $2$
Conductor $6840$
Sign $0.583 + 0.811i$
Analytic cond. $54.6176$
Root an. cond. $7.39037$
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  + i·5-s + 1.73·7-s − 4.37i·11-s + 4.08i·13-s + 2.47i·17-s + (−4.12 − 1.42i)19-s − 3.66i·23-s − 25-s + 1.18·29-s + 0.741i·31-s + 1.73i·35-s − 8.33i·37-s + 6.48·41-s − 12.2·43-s − 6.85i·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + 0.653·7-s − 1.31i·11-s + 1.13i·13-s + 0.600i·17-s + (−0.945 − 0.325i)19-s − 0.763i·23-s − 0.200·25-s + 0.219·29-s + 0.133i·31-s + 0.292i·35-s − 1.37i·37-s + 1.01·41-s − 1.87·43-s − 0.999i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6840\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.583 + 0.811i$
Analytic conductor: \(54.6176\)
Root analytic conductor: \(7.39037\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6840} (3761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6840,\ (\ :1/2),\ 0.583 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.760897635\)
\(L(\frac12)\) \(\approx\) \(1.760897635\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - iT \)
19 \( 1 + (4.12 + 1.42i)T \)
good7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 + 4.37iT - 11T^{2} \)
13 \( 1 - 4.08iT - 13T^{2} \)
17 \( 1 - 2.47iT - 17T^{2} \)
23 \( 1 + 3.66iT - 23T^{2} \)
29 \( 1 - 1.18T + 29T^{2} \)
31 \( 1 - 0.741iT - 31T^{2} \)
37 \( 1 + 8.33iT - 37T^{2} \)
41 \( 1 - 6.48T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + 6.85iT - 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 6.46T + 61T^{2} \)
67 \( 1 + 7.60iT - 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 - 1.43iT - 83T^{2} \)
89 \( 1 + 0.714T + 89T^{2} \)
97 \( 1 + 2.31iT - 97T^{2} \)
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

−7.962275776032756860951289871345, −7.04211831116452288797916365999, −6.49713017129178824139861052150, −5.84868056028047262111570987533, −5.01281549918572778642900598620, −4.15451971563374694179429727295, −3.58192753569016154146715826206, −2.48259277534832313512808313640, −1.79683268767689483714862425286, −0.48306206918415975065508991326, 0.986993976132878613721730368171, 1.91414120330302519509839550189, 2.76080182898155409313247661585, 3.83271686425599718986432151191, 4.61445119076848546936204442202, 5.13627033483649455316074974582, 5.83034374552117563833686411506, 6.81459505316331249563702340770, 7.40280434125160499461953482974, 8.175289316642091937953970079970

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